STELLINGEN

1. Als A een eindige verzameling is bestaande uit n elementen, met n een

even getal, en a een familie van n deelverzamelingen van A die ieder uit

een even aantal elementen bestaan, dan bevat a (minstens) twee verzame-

lingen waarvan de doorsnede uit een even aantal elementen bestaat.

2. Voor een Isingraodel met willekeurige n-spininteracties geldt, voor wille-

keurige verzamelingen spins A, 3 en C, de volgende correlatiefunctie-

ongelijkheid:

% > 2 lDCC DCC

waarin, voor iedere verzameling spins X, a = I", c .A .r vvfcX

3- Ten onrechte suggereren Hirsch en Smale dat het stelsel differentiaal-

vergelijkingen:

x = (A-By-Ax)x; y = (Cx-D-py)y ,

niet A, B, C, D, X en v groter dan nul, een limietcykel zou kunnen hebeen.

M.W. Hirsch en S. Smale

Differential equations, dynamical systems

and linear algebra.

'•>. Voor een Isingmodel gedefinieerd op eer. oneindige planaire graaf in een

homogeen magneetveld ':i geldt voor een randrij (v

getal, de volgende correlatiefunctie-identiteic:

homogeen magneetveld ':i geldt voor een randrij (v , ...,v ), met n een even

/ • I l n ilim J y (-i)1^!. H i n o > V = 0.

5- -tet een eenvoudig voorbeeld kan men laten zien, dat de door Lyons et al.

noodzakelijk geachte uitbreiding van het theorema van Faxen voor het

geval van niet-lokale visecóiteit fout is.

K.B.- Lyons, R.C. Mockler and W.J. O'SuDivan

Phys.Rev. A10 (1972) 393.

/

• . l.aa-, :-. -i- :.amilto::: aar, z i j : ; var. een isi.'igmodei ir.ct wi l lekeur ige n-sp in-

lr.',er*i'V : • 0 on !.'(}!; du 2 1 ' 1 1 •! J , r'.3trix r.et r iatr ixelenenten

:,, L(:;) = 2V_ * ' <a,op>H , v.'aarin V de verzameling væi a l l e spins i s ,

A,BCV •.•n Z de kan'.nieke to'-standSKor., dan ge ld t :

•J(-:i) = C(H)"1 .

7. j ' o r reuent zijr. <U.or •.•.•s.;-.! -=t n i . nauÆeurip. : -T :i-:en ingen gedaan aan

ï'-- oa : : f e s t ruc tuu r va:. .'r..J>:\ 'n d<. A-V. faso. ïv-r.eir.de him "berekenir.gei.

Le :iur,n-rn ver£ iS- i J •'-'-!'. ™t -:•.;." r I.'.T-::! •.-lo gfff^v^ns van !!L,3n in de super-

re lo i ' ] ' !id' '. O' 'jtijnJ, i j !v.-t .:j.rivol d~ invloed tü : r.'Ti'il"i- van de vcran-

d^rlr.j; va:, do KrJ .;tal:j t r u f f i u r , ••.:•:.£: volg< van d-- s t r u c t u r e l e fase-

:.'Vc !'('"•!:.(3, 01 dr i andonc ^ruo'Miur.

Å.T. •'••.;.;••!, ::.".•.'. '-'.yrur: ei. F.'". Vuol ler , preprint . .

^. !:ot : J •eenvoudiger o:r. .io "- - --a:.!:: '..'iiv-.1 va:, do *.'jt ::u toe opgeloste

ij-v.'r:.o>3:odL'liori -jp -lot /.wa'irat i 3':-. roos te r v-_ Lerekener. raet benulp van

do overdrac:itr:mar,r i>: dir- ::t.-t rouj tor diagcr.aal &pL,ouvt dan .-net de over-

Jracr.Ljmacrix :lie nc-t rocõL-~r r i j v>cr r i j opbouwt.

Laa:, v-; i r een IK ir.gr-:cdc !. n:o*. -..•li;-,-—irlfx- r i-bpir . ir . teracties , H dat

J'-__ var. de iiaaiitor.I&a:' - . i j ; . , dai !~ ".íiacelverKiug van een gegeven spin

v. :r.'.j'. d.; overige so'.i.s IJ -;_•'•.rij:" . V.-si. eenvoudig argument l a a t zien

dat vorr alit- v rzze-.^.l'.r.S'-"' ^ ; ' : : •' . '- . v ;.] '. -rva-V-.-ri ge ld t :

'",.> = <J- / ' ~'~> • "->'' ir. de . l -r-r . j ' . r 7 .-rn-"-j.d'; r e f e r e n t i e s afgeleide

; t í . : . i - j : l i n e a i r e r e i a t : « j :u.-.;':ii d-; Jpinuorrelaxief ' -mcties zi jn equi-

valent .-net Pover.staanae r^l; Jr , le r: en dus rr.ci e lkaar .

R. :,--.-:.-,Vi-jr on J . Xogi'^-o, P\:jiicn •',.-' {V>1?) 23 -

J. \ e r t e t ; en en ••..?,. de Vri - s , Corir..' •%•.',.;••!.%•-.. ."1 {^J^^} 131.

: . ~jru: ..-r e,. C. •.•'.•_-rLi:1l, ?i:y^lca -'T (1C'73) 303.

10. lier. v e r c c n i l da: r r i .eataat iuisc:i net medische r.ccicl en het s o c i o l o g i -

rcr.e .-aodol voor "aiVlJK^nd" .;;-.drag v.Ttoo.-t uen s t e r k e overeenkomst met

dat tussen i d e a l e inodeller: «:. >ii e^ - ldea lo rci.dellen A'oor fys i sche s y s t e -

men. Op grond hi^rvar. kal. m=n verrr.C" der. dat het medische model s l e c h t s

o"r; naldo-orde benadering i s .

/

er^rcuven Jr. Je analogie russer. ã<_.

va:. r°r.---x;:i'-:-.--.3it r rc :a r ; /o ten ''-n ii--- m SUL1 ar;,1

do~r

var. -"'ucaryotor

í-autr., ProgT. i:: ."i'uül. Aciá & Mol.Ijicí 1. i

J i l b e r c , Mature 'J'f1 ( I ivó) ' 0 1 .

ON CERTAIN RELATIONS BETWEENSPIN CORRELATION FUNCTIONS

OF ISING MODELS

PROEFSCHRIFT

ter verkrijging van de graad van doctor inde wiskunde en natuurwetenschappen aan de

Rijksuniversiteit te Leiden, op gezag vande Rector Magnificus Dr. D. J. Kuenen,

hoogleraar in de faculteit der Wiskunde enNatuurwetenschappen, volgens besluit vanhet college van dekanen te verdedigen op

woensdag 14 juni 1978te klokke 15.15 uur

door

RICHARD JOHN BOEL

geboren te Rotterdam in 1951

Krips Repro — Meppel

Promotor: Prof.dr. P.W. Kasteleyn

Aan hen die dit. proefschrift lezen, maar vooral

aan hen die het graag zouden willen kunnen lezen

CONTENTS

INTRODUCTION 7

I. CORRELATION-FUNCTION IDENTITIES FOR GENERAL PLAIIAR ISING SYSTEMS 11*

1. Introduction 15

2. Definitions and formulation of the main theorem l6

3. Graph-theoretical preliminaries 18

h. Proof of the main theorem 21

5. Corollaries and some applications 27

6. Concluding remarks 31

References 32

II. CORRELATION-FUNCTION IDENTITIES AND INEQUALITIES FOR ISING MODELS

WITH PAIR INTERACTIONS 3^

1. Introduction 3^

2. Conditions for the existence of certain identities and

inequalities for correlation functions 35

3. Generalizations of Theorems 1 and 2 h'y

h. Some properties of sets of A-idsntities kj

5. Examples of A-identities 50

6. Examples of A-inequalities 5I4

7. Concluding remarks 56

References 57

III. EXTREMAL A-INEQUALITIES FOR ISIHG MODELS WITH PAIR INTERACTIONS 58

1. Introduction 58

2. Definitions and notation 58

3. Polyhedral convex cones 60

It. Extremal A-inequalities 61

5. Examples of extremal A-inequalities 65

6. References 70

IV. COPRELATION-FUflCTION IDENTITIES FOR GENERAL ISIIJG MODELS 71

1. Introduction

2. Definitions and notation

3. General formalism

h. Applications

5. Concluding remarks

References

72

72

lh79

85

86

SAMENVATTING 87

Ernst Ising showed a remarkable form of foresight when in 1925 he wrote:

"Trotadem dürfte die vorliegende Untersuchung für das Problem des

Ferromagnetismus von einem gewissen Interesse sein."

E. Ising, Z.Phys. 31 (1925) 253.

INTRODUCTION.

One of the main fields of research in experimental and theoretical

physics of condensed matter in thermal equilibrium is that of the study of

phase transitions. A phase transition is characterized by the fact that the

number and nature of thermal equilibrium states of a system are changed by

smp.ll variations of externally controlled variables. Associated with such a

transition is a drastic change in the values of certain macroscopic

variables of the system.

As a typical example we consider magnetic crystals (of which certain

aspects will form the subject of this thesis), which consist of a large

number of particles, occupying the sites of a crystal lattice, each

particle having a spin and a corresponding magnetic moment. For certain

types of these magnetic crystals (ferromagnets) there is a temperature above

which they do not show a magnetization and below which they do (spontaneous

magnetization). Associated with this (critical) temperature is a divergence

in the values of the susceptibility, and sometimes the specific heat, at

zero magnetic field.

The aim of equilibrium statistical mechanics is to give a theoretical

description, on the basis of the microscopic structure of the system, of

such phenomena. Since a phase transition is due to co-operative behaviour

of large groups of interacting particles, the interactions between these

particles play a crucial role. This implies that one cai?not describe such

systems as nearly ideal (i.e. as deviating from ideal systems by the

presence of a weak perturbation), which complicates the theoretical treat-

ment considerably.

The first and simplest model proposed to describe the behaviour of a

magnetic crystal is the Ising model (for the history of which the reader is

referred to a review article by Brush ). The model consists of a set of

vertices or points (representing the particles of the crystal), and with

each vertex v is associated a spin variable a (representing the spin of

the particle at v ) , which can take the values +1 or -1. The spins inter-

act pairwise with each other, and the interaction energy of a pair of spins

where J , is a real variable representing the strength ofis -J ,00,w ' v v'

the interaction. If J , > 0 , the interaction is called ferromagnetic and

the energy is minimal of both spin variables are equal; if J ,< 0, the

interaction is called antiferromagnetic and the energy is minimal if the

spin variables have opposite values. If the crystal is placed in an

external magnetic field, a spin will in addition have an interaction with

this field with interaction energy -mHc_, where m is the magnetic moment of

the spin and H the magnetic field. The total energy of the system is thus

given by

H = - ä I J ,a a , - mH J aw' v v ^ v

2)

where the first sum is over all pairs of vertices, of which the associated

spins have an interaction,and the second sum is over all vertices. If one

assumes that the system is in thermal equilibrium at temperature T, then, by

the standard method; of equilibrium statistical mechanics, the thermo-

dynamic functions such as the frt'.p energy, the energy, the entropy and the

specific heat can be obtained from the (canonical) partition function.

A major breakthrough in the study of this model is due to Onsager

who succeeded in calculating exactly the partition function of the two-

dimensional Ising model on a square lattice in zero magnetic field. He

also gave an expression for the magnetization. It turned out that the model

exhibits a phase transition, but that the nature of the transition is quite

different from what existing theories had so far predicted. Until now

nobody has succeeded in deriving expressions for the partition function of

the two-dimensional Ising model in a field or the three-dimensional model.

At first sight the model seems highly artificial (it was considered as

such for a long time) and the results for the two-dimensional case seem

to be only of theoretical interest. However, experimental physicists have

succeeded in finding compounds in which the interactions between the spins

of the particles have a two-dimensional character and can be described by an

Ising-like interaction. The experimental results on the specific heat and

magnetization of these compounds beautifully confirm the results obtained by

Onsager. For details on experimental results, we refer to a review

article of De Jongh and Miedema or the Proceedings of the I976 Inter-im

national Conference on Magnetism

However, it is not only the successful description of a me.gnetic crystal

which makes the Ising model interesting. The model can be considered as

describing any system of interacting units, which can be in two different

states. These two states can, for instance, be interpreted as corresponding

to the presence or absence of a particle at a vertex, and thus one has a

model for a lattice gas. Furthermore, it can be used in the case that

every vertex can be occupied by two different kinds ol' atoms, and one has

a model for a binary alloy. For some applications of the Ising model to

biology (for example the helix-coil transition in D1IA) we refer the reader

to ref. 5- Recently, the Ising model has also found applications in6 7)

quantum field theory '

Although the thermodynamic functions are very important in the

description of the model, there is another set of quantities which are of

fundamental importance, nanely the correlation functions. Correlatie:

functions are thermal expectation values of products of spin variables, and

are a measure for the way the spins at different vertices are correlated,

and hence for the order and symmetry present in the system. They give a

more detailed description of the microscopic properties of the model.

Furihermore, the spontaneous magnetization and the magnetic susceptibility

in zero magnetic field can be expressed in terms of correlation functions.

Since no explicit results are known about thermodynamic functions in case a

magnetic field is present, the correlation functions can give important

information about the magnetization and susceptibility . The

susceptibility is also related to cross-sections in neutron scattering on

magnetic systems and the correlation functions are important for the

interpretation of this kind of experiments.

The results which are known at the moment about correlation functions

in Ising models can be divided into two classes. First, there are explicit

expressions for the correlation functions. This is the case for the one-

dimensional model, which is, however, not very interesting, because it does

not exhibit a phase transition. Results for certain correlation functions

for some specific lattices (square, honeycomb and triangular) are also9)known. Very recently, McCoy, Tracy and Viu gave closed expressions for

all correlation functions of the two-dimensional model on a square lattice

in zero magnetic field. This exhausts the results thus far obtained in this

class. Secondly, there are general rigorous results about correlation

functions, which again fall into two classes, namely algebraic equations for

correlation functions and algebraic inequalities for correlation functions.

Exact algebraic equations for the correlation functions are, for example,

the Kirkwood-Salsburg equations11)

10)and analogous sets of equations due to

12)Gallavotti and Miracle-Sole and to Gruber and Merlini ""'. These are

linear inhomogeneous equations for the correlation functions in which the

/

coefficients depend explicitly on the interaction parameters of the system.

Algebraic equations for correlation functions not explicitly containing the

interaction parameters of the system, to be called correlation-function

identities, were until now known only for the one-dimensional Ising model

and for the Ising model on a tree

Algebraic correlation-function inequalities are statements about the

sign of certain algebraic functions of correlation functions. For example,

in ferromagnetic systems (in which all interaction parameters are non-

negative) the spins tend to align parallel to one another, since this lowers

the energy. This tendency is reflected in the statement that the pair-

correlation function is positive, which ws.s first proved by Griffiths and is

known as the first Griffiths inequality. This inequality is one of a set of1U)

inequalities known as the GKS inequalities , which are due to Griffiths,

Kelly and Sherman. Another inequality for ferromagnetic systems was derived

by Griffiths, Hurst and ShermanTt) (and is known as the GHS inequality); it

proves the concavity of the magnetization as function of the magnetic field.

15) Percus16)

Further known inequalities are due to, for example, Lebowitz

Newman and Sylvester

Both the algebraic equations and the algebraic inequalities have been

applied to prove the existence of the thermodynamic limit of correlation

functions, the existence or absence of phase transitions, and the

analyticity and monotonicity of thermodynamic functions, to derive bounds

for correlation functions in terms of lower-order correlation functions (i.e.

correlation functions containing fewer spin variables) and to give estimates

of critical temperatures. For a general reference, see for instance

Ruelle and Lebowitz . Furthermore, since some of the inequalities

can be generalized to higher-spin systems and continuous-spin systems they

have found extensive applications in quantum field theory ' . So in the

absence of explicit results these algebraic equations and inequalities can

be a powerful tool for obtaining information about the system.

In this thesis we shall study certain relations between spin correlation

functions in Ising models. Let A be a set of vertices and, if B is a subset

of A, let A\B denote the set of vertices in A but not in B. By a, we denote

the product of all spin variables associated with the vertices of A and by

<aA> the thermal average of o.. We shall study the family of functions

10

where the sum is over all subsets B of A, and where X is, for all BcA an

arbitrary complex number which does not depend on the interaction parameters

of the system; some generalizations of these functions will also be

considered. The main object of this thesis is to find conditions for the

X_ under which A for a given Ising model (specified by a set of vertices

and a set of pairs of spins which have an interaction) is

(a) equal to zero for all possible choices of the values of the interaction

parameters

or

(b) nonnegative for all possible choices of nonnegative values of the

interaction parameters.

The family of functions A„ for which (a) applies then determines a family of

correlation-function identities, and the family of functions A for which

(b) applies a family of correlation-function inequalities.

In chapters I, II and III the Ising models which we consider contain

only pair interactions. By representing the interaction between two spins

by an edge between the corresponding vertices we can, in a natural way,

associate with each Ising model a graph (and speak of an Ising model defined

on a graph). The coupling parameters associated with the interactions then

define a function on the set of edges. The graph-theoretical concepts and

lemmas developed and proved, respectively, in chapter I will be used

throughout the thesis.

In chapter I we shall give an example of a family of functions A.

(specified by the choice of A) for which (a) applies. The Ising models

considered will be Ising models on planar graphs.

In chapter II, (a) and (b) will be studied for Ising models on arbitrary

graphs. A necessary and sufficient condition on the A_ will be derivedü

under which (a) or (b) applies. It will turn out that in case (a) theseconditions consist in a set of homogeneous linear equations in the A

a

with coefficients 0 or 1, and in case (b) in the corresponding set of

homogeneous linear inequalities. The coefficients 0 or 1 are closely

related to the way in which the vertices of a given set A are connected

by edges of the graph, and can easily be determined. Examples of (a) and

(b) will be given.

11

Chapter III vill contain a study of the linear inequalities obtained in

the preceding chapter, and results from the theory of linear inequalities.

The main conclusion will be that for a given set A, any correlation-

function inequality of the form \ i ° can be written as a positive linear

combination of a fixed finite set of "extremal" correlation-function

inequalities of this form. Examples of extremal inequalities will be given

for the cases that A contains h and 6 vertices. Furthermore, some extremal

inequalities valid for general A will be derived. For Ising models with

pair interactions only, these extremal inequalities will give the best

possible upper and lower bounds of correlation functions in terms of lower-

order ones within the class of inequalities considered.

In chapter IV, the functions A. for which (a) holds will be studied in

the case where general n-spin interactions are present in the system. It is

proven that every A for which (a) holds can be written as a linear

combination of simpler functions (namely functions A for which A takes the

values 1, -1 or 0 for all BCA), each of which satisfies (a). The study of

these simpler functions turns out to be rather simple, and a necessary and

sufficient condition will be derived under which (a) holds. Again, some

examples will be given.

References.

1) S.G. Brush, Rev.Mod.Phys. 32(1967) 883.

2) L. Onsager, Phys.Rev. §5_ (l^h) 117-

3) L.J. de Jongh and A.R. Miedema, Adv.Phys. .23 ( W * ) 1.

't) Proceedings of the International Conference on Magnetism, Physica (B+C)86-88 (1977).

5) C.J. Thompson, Mathematical S t a t i s t i ca l Mechanics (Macmillan, New York,1972).

6)

7)

8)

B. Simon, The P( <j>)_ Euclidean (Quantum) Field Theory (PrincetonUniversity Press, Princeton, ^)

9)

Constructive Quantum Field Theory, G. Velo and A.S. Wightman eds.(Springer-Verlag, Berlin, 1973)-

D.B. Abraham, Phys.Lett. J59A (1972) 357,Phys.Lett. 6JA (1977) 271.

T.T. Wu, B.M. McCoy and C.A. Tracy, Phys.Rev. B1J3 (1976) 316.

B.M. McCoy, C.A. Tracy and T.T. Wu, Phys .Rev.Lett. 2s8 (1977) 793.

12

/

10) D. Ruelle, Statistical Mechanics (V/.A. Benjamin, New York, 1969) Ch. k.

11) G. Gallavotti and S. Miracle-Sole, Comm.Math.Phys . _5 ( 1961) 21 5 -

12) C. Gruber and D. Merlini, Physica 6j (1973) 308.

13) H. Falk, Phys.Rev. F[2 (1975) 518U.

lit) R.B. Griffiths, in: Phase Transitions and Critical Phenomena, Vol. 1,

C. Domb and M.S. Green eds. (Academic Press, London, 1972).

15) J.L. Lebowitz, Comm.Math.Phys. 21 C 19T1*) 87.

16) J.K. Percus, Comm.Math.Phys. J*0 (1975) 283.

17) CM. Newman, Z. Wahrscheinlichkeitstheorie verw. Gebiete 32 (1975) 75.

18) G.S. Sylvester, Comm.Math.Phys. kg (1975) 209.

19) J.L. Lebowitz, in International Symposium on Mathematical Problems in

Theoretical Physics, H. Araki ed. (Springer-Verlag, Berlin, 1975) p.370.

13

/

I. CORRELATION-FUNCTION IDENTITIES FOR GENERAL FLANAR ISING SYSTEMS

Abstract.

A system of exact algebraic relations is derived for the spin correlation

functions on any so-called "boundary set" of an Ising model on an arbitrary

planar graph. One way of expressing these relations is to say that every

higher-order correlation function is equal to a Pfaffian of the pair

correlation function on the same set of boundary spins.

/

1. Introduction.

In this paper we shall derive and study a set of identities which express

certain higher-order correlation functions of an arbitrary planar Ising model

(in zero magnetic field) in terms of the pair correlation function of this

model. These identities do not contain explicitly any interaction parameter

of the system and they hold for any values of the parameters.

As is well known, identities of this general nature, i.e. algebraic

relations between correlation functions of various orders not involving the

interaction parameters, occur in many branches of physics. Most often they

are used in order to close an otherwise infinite hierarchy of equations and

to obtain a finite set which then can be handled much more simply. For this

reason such relations are often called "closure relations".

One distinguishes between exact and approximate closure relations.

Perhaps the best known example of the latter is Kirkwood's superposition

approximation in the theory of classical fluids and its use, in two different

hierarchies, to obtain the (approximate) Kirkwood and Born-Green integral

equations, respectively, for the pair correlation function '.

Exact identities, on the other hand, are far more scarce and they are

only known for systems which, in a sense, are exactly solvable. We mention

the following examples:

i) The classical one-dimensional system of particles interacting via an

arbitrary nearest-neifc-ibour interaction in an arbitrary (inhomogeneous)2)

external potential

ii) It is well known, see e.g. ref. 3» that for an open Ising chain with

equal nearest-neighbour interactions in zero field every even-order

correlation between Ising spins factorizes into a product of pair correlations

each of which, in turn, is a product of nearest-neighbour correlations,

iii) Recently, Falk has generalized these relations to the case of the

zero-field Ising model on a Cayley tree.

The identities which we shall derive in the present paper can be

regarded as a generalization of Falk's result. This connection between the

two results will be elucidated in section 5, application 3(a).

The outline of the paper is as follows. In section 2 we define the main

«) This result can in a straightforward way be generalized so as to cover

also the case of an inhomogeneous Ising chain, i.e. an Ising chain with

arbitrary nearest-neighbour interactions,in an arbitrary field.

15

/

concepts to be used and state the main result (Theorem A ) . In section 3 some

graph -theoretical lemmas on planar graphs are proved. Section U contains

the proof of a theorem (Theorem B) which is shown to be equivalent to

Theorem A. The proof of Theorem B goes by simultaneous induction on two

variables, one of which is the number of edges. In section 5 we discuss

some corollaries and applications. We end the paper with a few concluding

remarks (section 6 ) .

The concepts and methods used in this paper are almost exclusively based

on abstract graph theory; only in one place (in the proof of Lemma 1) use is

made of topological graphs (i.e. of (plane) embeddings of graphs). Although

at first sight one would expect it to be simpler to use embeddings throughout

the analysis, it appears that the nonuniqueness of embeddings of graphs of

low connectivity (in particular disconnected graphs and graphs -'ith an

articulation vertex) would make our proof of Lemmas 2 and 3 more cumbersome;

to exclude such graphs is not possible because they may result from the

removal of an edge in the proof by induction. Lemma 1 can also be proved

without resorting to a plane embedding (viz. by using Mac Lane's

characterization of planar graphs ), but in that case it is the abstract

proof which is the more lengthy one.

2. Definitions and formulation of the main theorem.

We define a graph G to be a pair ;v(G), E ( G ) ) , where V(G) is a set of

elements called vertices and E(G) a set of unordered pairs iv,v'} of distinct

vertices, called edges. G is finite if V(G), and hence E(G), is finite.

For definitions of concepts used but not defined in this paper the reader is

referred to ref. 6.

In order to formulate the main result of this paper we introduce the

following two concepts.

First, let G be a planar graph, B a subset of V(G), and w a vertex not

in V(G). Out of G, B, and w we construct a new graph G_ with vertex set

V(G) U{w} and edge set E(G)UE B, where E ß = {{v,w} | V £ B } . We call the set

B a boundary set of G if G_ is planar.B

Second, let S = (v ,...,v ) be a sequence of (not necessarily distinct)

16

vertices of G, let w,w ,...,w (n >_1) be distinct vertices not in V(G), and

W the graph (called n-wheel for n>_3) defined by V('.V) = {w,w^,. .. ,w^},

E(w) = {{w,w }} if n=1 and E(W) ={{w,w.}, {w. ,w } | 1 £ j<_ n} if n > 2' J J d

(where w = w-). Out of G» S, and W we construct a new graph G with vertexn+1 i o

set V(G) UV(W) and edge set E(G) U E O U E( W), where E c = {{v. ,w.} I 1 < j < n} .o o j J —

We call the sequence S a "boundary sequence, of G if Gc is planar.If G is a finite planar graph, we shall understand by an Ising model on G

*)a spin system with pair interactions only, defined by the Hamiltonian

{u,v}eE(G)K a aUV U V

(1)

where o is the spin variable associated with the vertex u, assuming the

values ±1 only, K is the coupling parameter between the spins at u and v

(which is allowed to take complex values), and the sum is taken over all

edges of G. The unnormalized and normalized correlation functions (a»)G a n d

( r, ) _ are respectively defined by the equations

l V (2)(a ) = (o.) Z~ if Z ^ O,

where the summation is with respect to all spin variables of the system,

o. = II cr for any A C V ( G ) , and where Z, the canonical partition function,

is defined by Z (1) . For brevity we shall often omit reference to the

graph G and write (<?•) and ^a^-

Our main result states that each higher-order spin correlation function

on any boundary set B of G can be written as a Pfaffian of the pair

correlation function on B. More specifically:

Theorem A. If (v ,... sv ), with n >_1, is a boundary sequence of a finite

planar graph G, then for any Ising model on G with Z?0 the following identity

holds< a o . . . a > = P f C , (3)

V1 V2 vn

where C is the triangular array of elements given by:

W*) Factors kT have been absorbed into H and K .

In the above the Pfaffian is for n even defined by

... c p ,P

Pf C = I' e C cP P1P2

(he.)P P P1P2 P3P!* Pn-1 n

in which the sum is over all permutations P of (1, ..., n) with

P2i-1 < PSi " ^ - ^ '

and £ is the signature of P (cf., e.g., ref. 7); for n odd we define

Pf C = 0. (Ub)

3. Graph-theoretical preliminaries.

In the analysis we shall perform several operations on a graph G: the

deletion of an edge, the contraction of an edge {u,v} to a vertex u (or v),

the insertion of a vertex u' (u'^V(G)) into an edge {u,v} (being the addition

of the vertex u' to V(G) and the replacement of the edge {u,v} by the two

new edges {u,u'} and {u',v}), and the extension of G with an edge {u,v} with

u,v£V(G) (being the replacement of E(G) by E ( G ) U {{U,V}} ); note that in

the latter case the resulting graph is identical with G if {u,v} 6 EIG).

If u and v are distinct vertices of G, a chain between u and v is a

connected subgraph of G in which u and v have valency 1 and all other

vertices have valency 2. (Such a chain exists if and only if u and v belong

to the same component of G.) It is convenient to extend this definition by

calling the subgraph with vertex set {u} and empty edge set a chain between

u and u.

The following elementary properties of boundary sequences of any planar

graph G are used in the sequel:

(a) A subsequence of a boundary sequence is a boundary sequence.

(b) Any cyclic permutation of a boundary sequence is a boundary sequence.

(c) A boundary sequence S of G is also a boundary sequence of any subgraph of

G containing all vertices of S.

(d) The set of vertices in a boundary sequence is a boundary set of G. (For

a converse statement, see Corollary 1b of Lemma 1.)

18

The following property is used in some applications "but not in the proofs of

Theorems A and B:

(e) A subset B of V(G) is a boundary set if and only if, for some plane

embedding of G, there exists a face R of G such that all vertices of B

belong to the (topological) boundary of R.

This property is proven from the fact that two points of the plane lying out-

side a given open connected region can be joined by a curve lying entirely

inside that region if and only if they belong to the boundary of that region.

Lemma 1 • If v is a vertex of valency n^_2 of a finite planar graph G and U

is the set of vertices of G adjacent to v, then the vertices of U can be

ordered into a sequence (v. v ) such that the graph obtained by

extending G with the edges {v.,v. }, 1^.ii.n, where v = v ^ is planar.

Proof. Consider an embedding of G in the plane such that every edge is

represented by a straight line segment. Since G is finite we can draw a

circle C around v not enclosing any other vertex of G and not intersecting

any edge of G except the edges incident with v. Let w , ..., w be the

poini-í; of intersection of G and the latter edges, in an order in which they

are encountered when C is traversed. We consider the figure thus obtained

as a new (abstract) graph G'; for n^.3 the line segments into which C is

divided define n distinct edges, for n=2 they define one single edge. By

construction G' is planar. For 1 i<_n let v. be the vertex of V(G) adjacent

in G' to w. . If we now contract, for i £ i £ n , the edge {v.,w.} of G' to

the vertex v., the graph G" thus obtained is the extension of G with the

edges {v.,v. }, 1£i^.n. It can be shown that the contraction of an edge

in a planar graph results in a planar graph (cf. ref. 6, p. 61). It follows

that G" is planar. •

Corollary la. If v is a vertex of valency j 3 of a finite planar graph G and

u is a vertex adjacent to v, there exist two vertices u. and u„ in G,

distinct from each other and from u and adjacent to v, such that the graph

obtained by extending G with the edges {u,u.j} and {u,u2) is planar.

This corollary follows from Lemma 1 by identifying u with some v., and u. and

u_ with v. and v-+-|« respectively.

19

Corollary lo. The vertices of any boundary set B of a finite planar graph G

can be ordered into a "boundary sequence of G.

The case |B[ = 1 is trivial. For |B| >_ 2, the corollary follows from the

proof of Lemma 1, applied to the vertex w of G , if one puts S=(v v ),

G S=G'.

Lemma 2.**)

If (v.,Vp,v ,Vi ) is a boundary sequence of a planar graph G,

every cha:'n between v.. and v, has a vertex in common with every chain between

vo and v, . In particular if v =v , every chain between v and v, -must

contain v.; if further v p = v V a 1 1 f o u r vertices coincide.

Proof. Suppose the lemma is false. Then there exists a chain C between

v1 and v and a chain C , between v and v. which have no vei tex in common.

If we then form the graph G„, where S = (v ,v ,v,»v,), with the aid of a

1*-wheel W as introduced in section 2, the subgraph of G which is the uniono

of C ,, C i and W, extended with the four edges {v. ,w. } (i<_i <_!*), is

'nomeomorphic to the graph K_. By Kuratowski's theorem this is impossible

since G is planar (ref. 6, p. 6i). This proves the lemma. •

If G is a finite connected planar graph and (v , ..., v ) , withLemma

n >_3, is a boundary sequence of G, then:

(i) (vi» •••> v »v ) is a boundary sequence of G;

(ii) if v

and (v^

v , there exists a vertex v'GV(G) such that {v ,V'}SE(G)n n

, v ,v') is a boundary sequence of G.

Proof, (i) It is easily verified that, since G is connected, it contains a

vertex u (not necessarily distinct from v ,v or v.) such that there exist

three chains, one between u and vn-1'

one between u en v , and one betweenn

u and v , which have as their intersection the vertex u.

Consider the (planar) graph G , with S = (v , ..., v ), the (planar)

graph G' obtained from G„ by inserting a vertex w , into the edge {w ,w,},o n+i n 1

*) For any set B the number of elements in B is denoted by |B|.

**) This lemma has frequently been used in the theory of graphs, e.g. in the

theory of vertex colourings of graphs (see, e.g., ref. 6, p. 86).

20

and the graph G. obtained by extending G' with an edge {w + 1 , w . } , where

i2 <_jf_n-1. There exist in G. nine chains, one between each of the verticesJ

w ,s w ; w 01 the one hand, and each of the vertices w, w , u on the

other hand, such that any two chains have at most an end vertex in common.

1'nerefore G . contains a subgraph homeomorphic to the graph KJ

nonplanar by Kuratowski's theorem.

and hence is

VJe now consider the four vertices adjacent to w ,

and vn

By Corollary la of Lemma 1, applied to the vertices w and w

n+1

n " •" n+1'

and another vertexwe can extend G with two edges, each one between w

adjacent to w , so that the graph thus obtained is planar. These vertices

must be w and v s since w is excluded by the argument given above that

the graph G is nonplanar. The graph obtained from G' by extending it

with the edges iwn+i»

wJ a n d ' wn+i'

vn^ "^ J

ust tile SraPh G,,, corresponding

to the sequence S' = (v, v ,v ). Since G , is planar, S' is a boundaryi n n !

sequence of G.

(ii) Hext we consider the set of vertices adjacent to v in G', i.e. the set

{v|vev(G), {v,vn)eE(G)} U{w. |2 <_j <_n, v.=vn} . By Corollary 1a of Lemma 1,

applied to the vertices v and w , we can extend G . (defined under (a))n n+1 o

with two edges, each one between w and some other vertex adjacent to v ,

so that the graph thus obtained is planar. The edge {w ,w } cannot be oneof these edges because v1 j4 v , so that w is not adjacent to Nor can

either of these edges be of the form iwn+1>

w-^ with 2 <_j£n-1, again for the

reason that G. is nonplanar. The only remaining candidates are the edges

{w ,w } and {w i'v'} » where v' is some vertex in V ( G ) adjacent to v .

This shows that there exists at least one vertex v' adjacent to v in G

such that the graph G" obtained by extending 1 O 1 with the edge {w + ,v') is

planar. If finally we delete the auxiliary edge {w ,v }, the graph thus

obtained is just G „, where S" denotes the sequence (v. v , v ' ) . Since

G", and hence G „, is planar, S" is a boundary sequence. ^

h. Proof of the main theorem.

We define, quite generally, for any Ising model on a finite planar graph

G and any sequence of vertices (v , ..., v , v ) , with n^_0, of G the

21

following function:

, v ; v|G) = I (-Dj(ovov) (ov n c )j=1 j i=i i

v ; (5)

For brevity we shall in most cases omit reference to the ,;raph G in the

argument of f and write f(v., ..., v ; v) .

For later use we list the following simple properties of he correlation

functions (a.) and the function f:

(A) If G is the disjoint union of the graphs G' and G", and a and a are

products of spin variables referring only to the graphs G' and G",

respectively, then:

'YB'G' 'VO^B'G" •

(B) If n is odd and v-j ..., v are vertices of G, then

(C) f(v.,-..5V ;v) = 0 if n is zero or odd.

(D) If n^_2 and n is even, then

f(v1,...,vn;v) = -f(v n,v r..., V i;v) . (6)

(E) If n>_2 and v^ = v0, then

f(v1,...,vn;v) = f(v3,vlt,.. (7)

Since f is an entire analytic function in all coupling parameters K ,

where { U , V } 6 E ( G ) , we can make a convergent Taylor series expansion with

respect to all coupling parameters simultaneously of the form:

1 V v ) =m=0

where f (v.,...,v ;v) is a homogeneous polynomial of degree m in the

coupling parameters. This defines the functions f uniquely.

(8)

,v ,v,w)n

Lemma h. For any finite graph G and any sequence of \ertices (v,,.

of G with {v,w}EK(G) and n even, the functions f defined above satisfy,

for m > 1 :

22

/

3Kfm (V"V v ) = -f

Vie note that if G is planar and (v , ..., v , v, v) is a boundary sequence of

G, the arguments in the f-functions appearing in eq. (9) are, by Lemma 3(i)

and property (b), boundary sequences as well.

Proof. By a straightforward calculation, starting from eqs. (2) and (5),

using a = a = 1 and (-1) = 1 , we find

f(v1,...,vn;v) = f(v1,...,vn,v,w;v) + f(v1,....v^,v.wjw). (1O)

vw

Using eq. (6) we can rewrite this as

3

3Kvw

f(v1,...,vn;v) = -f(vf)v1,...,vn,v;v) + f( v1,... ,vn,v,w;w). (11)

Comparing terms of the same degree on both sides of this equation we obtain

eq. (9). ••

Theorem B. If G is a finite planar graph and (v., . .. ,v , v ) , with n>_1, a

boundary sequence of G, then for any Ising model on G

f(v1,...,vn; v|G) = 0, (12)

or equivalently, for all m>_0,

V v - ' - ' V viG) = c- (13)

Proof. Eqs. (12) and (13) hold trivially if n=0 or n is odd (property (C)).

Therefore, we may restrict ourselves to the case n^_2, n even. Further-

more it is sufficient to prove eqs. (12) and (13) only for the case v1 f v,

since the general case can be reduced to this case or to the case n=0 by

properties (D) and (E). So let v. f v.

We now prove eq. (13) for all m by induction with respect to the ordered

pairs of nonnegative integers (i-,m) where I = |E(G)|. We shall say that a

pair (Í.' ,m') precedes a pair (l,m) if £' <_ I, m'<_m and (I' ,m') ^ ( £,m). We

note that in the set of pairs (H,m) ordered in this way every element has

(at most) a finite number of predecessors.

We choose a particular fixed pair of nonnegative integers (£,m), a finite

23

/'

planar graph G with |iJCG) | = I, and a boundary sequence (v ,... ,v ,v) of G,

with n >_2, n even, and we take as our induction hypothesis that eq. (13)

holds for all cases labelled by a pair (ü.',m') preceding (£,m)

There are two possibilities to be considered: either G is connected or

it is disconnected.

First let C- be connected. Since by assumption v1 4 v we have

i = |E(G)j >. 1. If m=0, then f (v.,,. . . ,vn;v|o) = fQ(Vi,...,vn;v|GQ) where

G is the graph defined by V(G ) = V(G), E(G0)

= 0> i-e- t h e graph obtained

from G by deleting all its edges. Since (0,0) precedes (Ä,0) for i _* 1,

f (v ,...,v ;V]G ) vanishes by the induction hypothesis and so

f"0(v.|,...,v ;V|G) vanishes. Hence we may assume from now on m>_1.

Since I 1 and G is connected, v is not an isolated vertex of G.

According to Lemma 3(ii) there exists a vertex w€v(G) such that (v,w)EE(5)

and {v v ,v,w} is a boundary sequence of G (remember that we can assume

n > 2 and v ? v).I

We regard f (v.,...,v ;V|G) as a function of K . From eq. (9) togetherm l n vw

with the induction hypothesis we find that this function does not depend on

K so that we may restrict, ourselves to the case that K = 0 . But in thatvw vw

f m( V..., Vv|0) =

v*where G' is the graph obtained from G by deleting the edge {v,w} . Since

ISCO')! < |E(G)| the right-hand side of eq. (Ik) vanishes by the induction

hypothesis. This proves the validity of eq. (13) for all cases labelled

(£,m) where the graph G is connected.

Suppose now that G is not connected, and let G' be the component of G

containing v. Let J = {j I v. ey(G' ), 1 < j <n} and J' = {1,...,n} \J. The

terms of the sum in the right-hand side of eq. (5) with j €J' are zero

because (0 .0. ) factorizes into vanishing factors (see properties (A) and

(B)). The other terms (with j ej) also factorize and we obtain:

itj 1(15)

*) 111 is method of induction can easily be reduced to the ordinary method of

induction with respect to the number of predecessors of an element. See also

ref. 8.

2k

where 0" is defined by V(G") = V(G) W j G 1 ) , E(G") = E ( G ) \ E ( G ' ) .

If |j| = 0, tlie sura in (15) is zero, and if |j|, and hence |J'|> is

odd, the second factor of (15) vanishes; in both cases eq. (12) follows

trivially. Let now J = {j ,j ,...,j } with p ^ 2 , p even and

1 IJ i < i2 < . . . < j In, and let K = { i | j < i < j 2 } and L = J' \ K. For all

k i=K and tSL it follows from Lemma ?, applie 1 to the boundary sequence

(v. , v. , v. , v ), that v is not in the same component as v . Hence

(i"._T1a )_,, factorizes into two factors, one of which contains only thele:J V-^ Ci

spin variables o (k£K). This factor is zero, and hence eq. (12) holds, ifvk

M = j -j—j i —1 is odd, so that we may assume that j —j is odd. Similarly,

we can argue that j ,,-j can be assumed to be odd for every r (1 j^r^p-1).

In that case (-1) r = (-1)J1~ (-i)r. Introducing the notation

u = v. (1 <r<p) we can writer Jr ~ ~"

G) =(-1) (1)G„(iSI'

p

r=1

= (-1) (1L,,(G ,eJ,) „ f(u ,...,u ;v|G')

. G 1 p(16)

Since (u,,...,u ,v) is a subsequence of (v,,...,v ,v) it is a boundary1 p in

sequence of G (property (a) of section 3) and also of G' (property (c)).

It follows from eq. (16) that f (v ,...,v ;V|G) can be expressed

linearly in terms of f ,(u ,...,u ;V|G') with m' < m, where G' is connectedm l P

and |E(G')| £_E(G)| . Hence, from the induction hypothesis together witn the

above result for connected graphs, we deduce the validity of eq. (13) for

all cases labelled (£,m), and hence of eq. (13) and eq. (12) in all cases. •

In order to prove Theorem A we consider a boundary sequence (v.,...,v )

of a planar graph G.

The validity of eq. (3) for n odd is an immediate consequence of eq. C*b)

and property (B). NOW let n be even and n >_2. From Lemma 3(i) and property

(b) of section 3 we conclude that also (v.,...,v ,v ) is a boundary sequence

of G. Applying therefore Theorem B, eq. (12), to this case we obtain, using

eq. (5) and dividing out a factor Z (which is non-zero by assumption):

<ov > =j=2

(-1)J <o > < n ai=2 x

(17)

This is, for n even, the well-known expansion formula for a Pfaffian of a7)triangular array with respect to the elements of its first row , from

which eq. (3) can be obtained by iteration. This completes the proof of the

theorem. •

We remark that, conversely, Theorem B for n even and Z^O follows

directly from eq. (17)> and hence from Theorem A. To show this we apply

eq. (17) to the second factor in the right-hand side of eq. (5). This yields

n n

j=1 k=1

d+k. nJli=1

(18)

where e . = sgn(k-j) if k^j and e,kj K

= 0 . Since e . is antisymmetric in k andkj

j, and the product of the other factors in the terms on the right-hand side

, ...,v ;v),

9)

of eq. (18) is symmetric, the double sum, and *\ence f(

vanishes. The validity of Theorem B for the case Z=0 follows from a

continuity argument. •

Eq. (3) shows a formal analogy with certain results of Green and Hurst

These authors showed, on the one hand, that their S-matrix method for

calculating partition functions of a class of lattice models (now called

vertex models) can be applied to planar lattice graphs only if the

coefficients in a certain operator polynomial which generates the high-

temperature series expansion of the partition function satisfy certain

consistency conditions.

Although a general form of these conditions is not explicitly formulated,

the examples given show that they are formally identical to the identities

(3) (ref. 9, p. 1o3 ff; ref. 10).

On the other hand, Green and Hurst proved that these consistency

conditions are satisfied for "effective" lattice models obtained from an

Ising model on a planar graph by summing over the interior states of certain

"decorating" subgraphs (ref. 9, p. 2\k ff).

The analogy is not accidental. To see this we observe that in the latter

case the coefficients in the operator polynomial can be interpreted, apart

from a common factor, as correlation functions of the boundary spins of the

*) There are some annoying misprints in ref. 9> e.g. in eqs. (U.36), (k.kO)

and {k. 1*1); cf. eqs. (20), (25), (26), and the following three lines of ref. 10.

26

decorating subgraph, considered as a separate Ising model. However, in the

present work we adopt a different point of view and we employ methods which

are entirely different from those of Green and Hurst.

Firstly, our emphasis is on arbitrary planar graphs and the existence of

correlation-funetion identities as such and not especially with a view to the

solution of lattice models via a decoration transformation; for a discussion

of the latter problem, see section 5, application 1. Secondly, the proof

of Theorem A is essentially graph-theoretical and more or less self-contained,

whereas the derivation of the analogous result by Green and Hurst requires

the full apparatus of their S-matrix method. Finally, there are a number of

differences of a more technical nature.

5. Corollaries and some applications.

The identities (3) and (12) have been derived here under the sole

restriction on the coupling parameters K that Z^O; otherwise all coupling

parameters are allowed to assume arbitrary complex values. For the sake of

completeness we also consider the case Z=0. By a simple continuity argument

one deduces from Theorem A the following corollary.

Corollary I. If Z=0, n is even, n>_l*, and (v , ...,v ) is a boundary

sequence of a finite planar graph G, then

Pf D = 0, where D^ . = (a ) » 1 li£.J£n-

Another extension of Theorem A is to the case of infinite graphs. For

such graphs neither the partition function nor the unnormalized correlation

functions are defined. Nevertheless, as is well known, one still can in

these cases attach a meaning to normalized correlation functions; this can

be done in various ways. One way is to define a correlation function <o. >A Lr

as the limit, for n •+», of the sequence < a. ) , where the G (n=1,2,...)R "n n

are finite subgraphs of the infinite graph G and ACV(G ) for all n.n

Theorem A now has the following consequence .

Corollary II. If B is any finite boundary set of an infinite planar graph G,

and (G^ G^,...) a sequence of finite subgraphs of G each one containing the

27

set of vertices B, and if for each pair of vertices u, v of B the limit

exists, then for any A C B the corresponding< o o > = limu v G n

limit (o. > = lim

< o o >u v Gn

< 0A>Gexists, and these functions satisfy the

identities given by eq. (3).

Applications.

1. The first application of the identities (3) concerns the question of the

solvability of eight-vertex models by the reduction of such models to

equivalent planar Ising models.

It is well known that the partition function of the eight-vertex model on

a square-lattice graph is equal to that of an Ising model on a square lattice

with properly chosen two-spin and four-spin interactions between spins on

each (elementary) square of the lattice (ref. 11, p. 3^8). Also, the latter

partition function is equal (up to a known factor) to that of an Ising model

on a (not necessarily planar) graph, obtained from the square-lattice graph

by the addition of a set of new vertices and new edges with properly chosen

coupling parameters; in fact, one vertex and four edges suffice (see e.g.

ref. 12).

If the latter graph is planar, the corresponding partition functions can

be evaluated exactly by one of the standard methods. The question arises

whether by considering the most general planar graph one would be able to

solve some hitherto unsolved cases of the general eight-vertex model. A

question equivalent to this, but phrased in the "dual language", can be

considered to have been answered in the negative by Green and Hurst in the

work discussed in the previous section. The same conclusion is reached using

the identities of this paper, as we shall indicate.

Consider a planar graph G decorating a square of the square-lattice graph

and having only the four corner vertices in common with this graph. Eq. (3),

applied in G, considered as a planar graph by itself, to the four spins on

the corner vertices, implies a relation between the two-spin and four-spin

coupling parameters in the equivalent generalized Ising model. If in turn

we translate this relation in terms of the equivalent eight-vertex model, we

obtain in the usual notation (ref. 11, p. 3hj)

This is precisely the so-called free-fermion condition. Such a condition

applies to the vertex weights at each vertex of the lattice; hence the

28

•^ight-vertex model is by definition a free-fermion model (ref. 11, p.

1 ''i model, which was first considered by Fan and Wu as a special case

of the i isht-vertex model introduced by them , was recognized by these

authors to be equivalent (via the high-temperature expansion) to the most

general (planar) lattice model which can, at least in principle, be solved

by the S-matrix method of Green and Hurst.

Hence, the above-mentioned procedure for solving eight-vertex models by

reducing the partition function to that of a planar graph does not lead to

the solution of a new class of eight-vertex models. For vertex models on

other planar lattices the situation is similar.

Conversely, however, one may use the above method to reduce the

calculation of the partition function of certain complicated planar Ising

models to that of simple free-fermion models. For example, it is now a

simple matter to reduce the calculation of the partition function of the

Ising model on the "Union Jack" lattice to that of a homogeneous free-

fermion model on a square lattice.

2. It follows from property (C) that the four spins around a square of the

square-lattice graph form a boundary set, and this can be ordered into a

boundary sequence, .j vp> v , '- Hence the identity (3), applied

to this sequence, expresses the correlation function <o a a a > inT1 V 2 V3 Tl(

terms of the pair correlations for pairs of nearest neighbours and pairs of

diagonal neighbours. The latter are well known for the translation-invariant

square lattice. Similar applications can be made to other planar lattice

graphs.

3. A next application concerns outerplanar graphs. A graph is called

outerplanar if for some embedding all vertices lie in the boundary of one

single face. In the terminology of the present paper these are planar graphs

for which the set of all vertices constitutes a boundary set (cf. property

(e)). It follows from property (a), Corollary 1b and Theorem A, that for

any Ising model on an outerplanar graph all correlation functions can be

expressed as Pfaffians of the pair correlation function.

Examples of outerplanar graphs are: (a) (Cayley) trees, (b) broken n-

vheels (obtained from n-wheels by deleting one edge of the rim), (c) poly-

gons with non-crossing diagonals. We now discuss some of these cases in

more detail.

29

(a) Correlation-function identities for Ising systems on a tree have been

obtained by Falk as mentioned in the intrc luction . Falk's second

decomposition theorem can be derived from the identities (3) or (17), applied

to a tree,by using the fact that in general a boundary set on a tree can in

more than one way be ordered into a boundary sequence. It is obtained by

applying eq. (I7) to each boundary sequence corresponding to a given set of

vertices, and (linearly) combining the resulting identities. Although this

derivation is, of course, not shorter than Falk's, it shows that his

identities are a consequence of (i) the identities (3) of this paper applied

to a tree, and (ii) the above-mentioned fact that a set of vertices of a

tree can be ordered in many different ways into a boundary sequence.

(b) An Ising model on a broken n-wheel is equivalent, via the usual dummy-

spin representation (see, e.g., ref. 15, p- 105) to an open Ising chain of n

spins in a magnetic field. The odd- (even-)order correlation functions of

the latter system are equal to correlation functions of the former system

which contain (do not contain) the spin variable associated with the dummy

vertex. It follows that for an Ising model with arbitrary nearest-neighbour

interactions in a (not necessarily homogeneous) magnetic field the identities

(3) remain valid for n even, whereas for n odd they are replaced by

<a •••0 > = Pf C + , (19)1 n

where C is the triangular array given by

ct.

( av.J

n a )v. v.1 J(c) Another case where Theorem A can be applied to any set of spins is that

of the open Ising chain with arbitrary nearest- and next-nearest-neighbour

interactions in zero field; it is easily verified that the graph

corresponding to this model is outerplanar.

h. As a last application we consider the conjecture that for ferromagnetic

Ising systems in zero field the (even-order) Ursell functions \J2i (defined as

usual) alternate in sign according to the formula

(-O*"1 u.,(i. W >- °

30

where i ,...,i label the spins. For i = 1,2, and 3 these inequalities are~ 17)known to be valid (cf., e.g., ref. 16). Also, Setô has shown them to

hold for trees for all i.

It can be shown , using the identities (3) of this paper, that the

above inequalities hold, for all H, for the spins on any boundary set B of

any planar (zero-field) Ising model with the property that every pair

correlation in this set B is nonnegative (e.g. for a ferromagnetic model,

according to Griffiths's first inequality). This generalizes Seto's result.

6. Concluding remarks.

The identities derived in this paper differ from other equations for

correlation functions of Ising systems such as the identities derived by19) 20)

Fisher and generalized by Dekeyser and Rogiers and by Gruber and21)

Merlini , or a hierarchy such as the Kirkwood-Salsburg equations (see,

e.g., ref. 22) in three respects: (1) they are non-linear; (2) they do not

contain the interaction parameters explicitly; (3) they are very special in

that their validity has been established only for planar Ising systems with

pair interactions (counterexamples for nonplanar systems are readily found).

There exists an interesting relationship between our results and some

results of Bedeaux et al derived for the one-dimensional Glauber model.

One of their results is that certain polynomials of time-dependent

correlation functions, called C-functions (which are analogous to the Ursell

functions except for the occurrence of minus signs in their definition),

vanish more rapidly for time going to infinity than the Ursell functions

themselves. Such C-functions can also be defined for the systems considered

in this paper. It is interesting to note that the identities (3), translated

in terms of these C-functions, are equivalent to the statement that all

C-functions of order n >_3 vanish identically for boundary sets of the

equilibrium Ising spin system considered in this paper (see also Kawasaki,

ref. 23, p. ^67 ff., where this situation is somewhat elucidated).

Throughout this paper we have restricted ourselves to relations between

spin correlation functions on boundary sets. It is possible to extend these

results and derive certain relations between correlation functions for more

31

general sets of spins in planar Ising models. This enables one in lurn to

derive relations between odd-spin correlation functions in infinite planar

graphs below the critical temperature. For these relations, and for a

connection of the identities derived in this paper with inequalities for

spin correlation functions we refer to subseqi. nt papers (ref. 2k).

References.

1) A. Munster, Statistical Thermodynamics, Vol. I (Springer-Verlag,Berlin-Heidelberg, 1969) p. 331 ff.

2) Z.W. Salsburg, R.W. Zwanzig, and J.G. Kirkwood, J. Chem. Phys. 2J_(1953) 1098.

3) D. Bedeaux, K.E. Shuler, and I. Oppenheim, J. Stat. Phys. 2_ (1970) 1.

M H. Falk, Phys. Rev. B12.(1975) 5181».

5) S. MacLane, Fund. Math. £8 (1937) 22.

6) R.J. Wilson, Introduction to Graph Theory (Oliver & Boyd, Edinburgh,1972).

7) E.R. Caianiello, Combinatorics and Renormalization in Quantum FieldTheory (W.A. Benjamin, New York, 1973).

8) G. Birkhoff, Lattice theory (Am. Math. Soc, Providence, 1967) p. I8I .

9) H.S. Green and C.A. Hurst, Order-disorder Phenomena (Interscience, I96U).

32

/

i:) C A . Hurst, J. Math. Phys. _5 (196M 90.

11) E.H. Lieb and ?.Y. V/u, in: Phase Transitions and Critical Phenomena,Vol. 1, C. Domb and M.S. Green eds. (Academic Press, London, 1972)pp. 332-U90.

1_') K. Jüngling and G. Oberraair, J.Phys. £7 ( U ï M L363.

13) C. Fan and F.Ï. Wu, Phys.Rev. B2 (1970) 7<-J3.

1h) V.G. Vaks, A.I. Larkin, and Yu.Il. Ovchinnikov, Zh.Eksp.Teor.Fiz. ]t_9[VjG'A 1i8u [ üov.Phys. JETP 22. (Vyuu) 8..J ] .

r ) r.Vf. Kasteleyn, in: Graph Theory and Theoretical Physics, F. iiararyed. (Acadenic Press, London, 19ú7) p. '<3.

lo) G.S. Sylvester, Comm. Math. Phys. u2_ (1,'T' ) .0.'.

17) II. S«tS, Progr. Theor. Phys. 5A (1975) 1881.

16) J. Groeneveld, to be published.

\j) M.E. Fisher, Phys.Rev. VVi (1959) 9É9-

Z'O) R. Dekeyser and J. Rogiers, Physica 53. (197^) 23-

21) C. Gruber andD. Merlini, P h y s i c a l (1973) 3O3.

_2) D. Ruelle, Statistict.l Mechanics (W.A. Benjamin, Ilew York, 19''9) p.80ff.

23) K. Kawasaki, in: Phase Transitions and Critical Phenomena, Vol 2,C. Domb and M.S. Green eds. (Academic Press, London, 1972) p. 1*66.

2l) R.J. Boel and P.V'. Kasteleyn, to be published.

33

l i . CORRELATIOlI-FUIICTIOiJ IDENTITIES A.7D IiJEQUALITIES

FOR ISING MODELS WITH PAIR INTERACTIONS

Abstract.

For Ising models with pair interactions in zero magnetic field a class

of linear combinations of products of two correlation functions is studied.

We derive sufficient and necessary conditions under which a function in

this class is (a) zero for all values of the coupling parameters, or (b)

nonnegative for all nonnegative values of the coupling parameters. Examples

of correlation-function identities and inequalities of this type are given.

1. Introduction.

1)In a recent paper '', to be referred to as I,it was proved that for a

(zero-field) Ising model on a planar graph the correlation functions for

spins on a so-called boundary set satisfy certain algebraic relations which

are valid for all values of the coupling parameters between the spins. These

relations can all be derived from a set of identities which can be written

in the form

I (-D J < a o > < ø a n aj=1 1 j 1 j k=1 V

> = o

where (v.,...,v ) is a sequence of (not necessarily distinct) vertices which

is a boundary sequence of the planar graph (cf. I eq. (17)).

The left-hand side of the above equation can be considered as a special

case of a function of the type E X < a > < o_a > , where G is an

arbitrary graph, A is an arbitrary set of vertices of G, D a subset of A,

the sum is over all subsets of A, and the coefficients AB are independent

of the coupling parameters; the case where all vertices in the boundary

sequence are different then corresponds to the case D=A, i.e. o„a_, = a.,,. .

/

In this paper we derive sufficient and necessary conditions on the

coefficients A under which a function of this general type for an Ising

model with pair interactions in zero magnetic field is (a) zero for all

values of the coupling parameters, or (b) nonnegative for all nonnegative

values of the coupling parameters, respectively.

In section 2 such conditions are derived for the case D=A on the basis of

an expansion of lhe Boltzmann factor with respect to the coupling parameters.

They take, respectively, the form of a set of linear equations and a set of

linear infaualities for the A with coefficients 0 and 1. In section 3 the

results are extended to the general case DC A. Section I4 is devoted to an

analysis of some properties of the sets of correlation-function identities

that can be obtained in this way. In section 5 various examples of such

identities are given, among which those derived in I. In section 6 we show

that some known correlatjon-function inequalities follow from the general

analysis given in this paper and we derive a new inequality. We end this

paper with a few concluding remarks.

Since the extension of the partition function and the correlation

functions of Ising models to complex values of the coupling parameters is

sometimes useful , we allow for these complex values where possible.

This implies that the partition function may take the value zero, in which

case the normalized correlation functions are not defined. For this reason

we shall work almost exclusively with unnormalized correlation functions ; the

translation of the results to normalized correlation functions is trivial.

2. Conditions for the existence of certain identities and inequalities for

correlation functions.

As in I we define a graph to be a pair (v(G), EIG)), where V(G) is a set

of elements called vertices and E(G) a set of unordered pairs {v,v'} of

distinct vertices, called edges. G is finite if V(G) and E(G) are finite.

For definitions used but not defined in this paper the reader is referred to

refs. 1 and 2.

35

/

An Ising model on a finite graph G is defined as a triple (G, /,K) , where

J is the set of all functions a: V(G) •+ {-1,1} (called configurations) and

K a complex function on E(G) (called the interaction function). The spin

variable o is the value of o at the vertex v, the coupling parameter K is

the value of K at the edge e. The set of all interaction functions will be

denoted r>y ?C, the set of all IC £ SC such that K >_0 (K > 0) for all ee£(G)

'oy 0? {f ) ; an Ising model (C,,S, K) with Í : E 7 ( ? + ) is called ferromagnetic

(strictly ferromagnetic).

For any set A C V ( G ) we define

a, = !1 a ; (1)

e A

for A =define

ea =

aX =

0 we

a av v

II

e ex

have o

r

e0

= 1. For any edge e = (v,v' } and any „ot. XCJl(G) we

The Hamiltonia.i of an Ising model (G,-/, K) is defined by

GSK(a) = -

e6E(G)

(2)

(3)

W

the unnormalized and normalized (spin) correlation functions (o.) and

<VG,K respectively, far any set ACV(G) by

-H„ „(a)

A'G.K

<a > = (a )r Z-1(5)

if

where Z, the canonical partition function, is defined bv Z = 'i)„ For

brevity, ve shall often-suppress the index K, and, where no confusion can

arise, the index G as well. We have taken 3=1.

Since the Hamiltonian is quadratic in the u , the correlation function

(a. )„ vanishes if |A| is odd. Therefore, we shall 'lenceforth consider

only correlation functions io.)r „ for even sets A, i.e. for sets with |A|

even.

We now consider an arbitrary even set A C V ( G ) . Let A = {A,,},^. be a set

of complex numbers defined for all even sets B CA, with the restriction

= A , for all B. We introduce the following quadratic combination of

36

/

ur.normlized correlation functions:

AA (G 'K) = (C)

where the sum is over all ever, sets BCA. For convenience, a function of the

type ((,') will i e referred to as a A-function. V.'e ahall nov.- derive a

condition for A under which a A-function satisficj the equation A CG,?'-) = 0

for all K £ 'A (to be referred to as a A-i dent i ty ), and a condition for A

under which it satisfies the inequality A (G,K/>_ 0 for all K - ? (to "be

referred to as a A-inequality)•

We first consider an arbitrary product of two unnormal]zed correlation

functions (a ) (o_),, with B, C C \ ' ( G ) , B n C = 0. If we expand the BoJtzmann

factors in this product with the aid of the elerr.entary relation

K CJ= C + S 0

e s

where e is an arbitrary edge, c = coshK and .; = sinhK^, we obtair

(7)

(8)

w h e r e ( o _ ) e t c . a r e c o r r e l a t i o n f u n c t i o n s o f t h e I s i n g m o d e l (G',f, K ' )B G

obtained from (G, </, K.) by deleting e from G and restricting K to £(G)\e

(which is equivalent to putting K = 0 in H„ .,(a) ).Using c = 1+s we rewrite eq. (8) as

G,K'

C'G B'G'

C8')

We nov; repeat tnis process for all other edges of G. To write the result

in a compact form we associate with each term in the resulting expression a

function

e : E(G) -> {0,1,2} , (9)

where ü^ = 0,1,2 labels the first, second and third term in the right-hand

side of eq. (8 1), respectively, and the edge sets L = {eeE(G))6e = 1} and

!•!„ = {eeE(G)|e = 2} . The set of all functions 0 is denoted by 9 . We

further define

g(e) = 1 1 g (9 ) , (10)eeE(G) e e

t- 37

where gg(0) = 1, gg( 1) =

r ( B , C )9 X C L Y Q i

g ( 2 ) = and

( 1 1 )

i n wh ich G-i i s t h e g r a p h d e f i n e d by V(G_,) = V ( G ) , E(G^)V 3 0

The r e s u l t o f a p p l y i n g e q . ( 7 ) t o a l l e d g e s o f G i s

( 1 2 )

A convenient way to characterize the various terms in r (B,C) is the0

following. We define for each 6 a graph G„ by V ( G J = V(G), E(Gj = l.U

i.e. 0 is obtained from G by deleting all edges e with 8 = 0 . For each

pair of sets XCL , ÏCH we define a function

: E(Ge) -> (13)

where Z i s t he f i e l d of i n t e g e r s modulo 2 , by

/' ( 1,0) i f eEX

(0 ,1 ) i f e € L \XO

(1,1) i f

( 0 , 0 ) i f eGM \Y

Conversely, every function of the type (13) with the property

(a + ß = 1 if e 6 L„

<Pe = (<*e>3e) witha + Ba = 0 i f eel-1

( 1 5 )

d e f i n e s two s e t s X C L , Y C M „ , and h e n c e a t e r m i n r . ( B , C ) , by

x = { e e L e | * e = (1 ,0 )} ,

Y = { e e M e U e = (1 ,1 )} .(16)

We remark that in the terminology of algebraic graph theory $ is a 1-chain

of Ge over Zg x Z£ (cf. ref. 3).

Let us now analyse the various terms of r (B,C). Since (a ) = Z o

and I _+ a = 0 , we have

(a) = 2IV(G)I,5. . for all ACv(G).

A G^ A,0(17)

38

Hence, the only nonvanishing terms in the right-hand side of eq. (11) are

those for1,which

X Y

VLe\X Y° = (18)

The first condition requires that every vertex in B is incident with an odd

number of edges of X UY and every vertex in V(G)\B with an even number of

edges of X UY; the second condition is analogous.

These conditions can easily be translated into a condition on the

function <(> if we introduce the functions 3<)>: V(G)->ZO*Z defined by

where the sum (taken in Z^ x Zo) is over all edges in Gg incident with the

vertex v, and x(B,C) : V(G) + Z 2 X Z, defined by

!

( 1,0) i f v S B

( 0 , 1 ) i f v e C (20)

( 0 , 0 ) i f v £ B u C .

3<t> and x(B,C) a r e 0 - c h a i n s of Gg ove r Z£ x Z j , 3<* i s c a l l e d t h e boundary of i>.

In t e r m s of t h e s e f u n c t i o n s , eq . (18 ) can be w r i t t e n a s

34> = x(B,C) . (21)

The s e t of a l l f u n c t i o n s if> w h i c h , f or a g iven c h o i c e of 8, B and C ( B n C = 0 ) ,

s a t i s f y e q s . (15 ) and (21) w i l l be d e n o t e d by * ( B , C ) . S i n c e e v e r y n o n -

v a n i s h i n g t e r m i n t h e r i g h t - h a n d s i d e of eq . (11) i s e q u a l t o 2C~

have

r . ( B , c ) = £

y?lV(G)I

(22)

We now proceed to derive a few properties of the sets 4>(B,C).a

We call a set S of edges of a graph G a (generalized) cycle of G if each

vertex of G is incident with an even number of edges of S. The total number

of cycles of G., including the empty cycle 0, will be denoted by YQ.

Lemma 1. If, for S e e and for disjoint sets B, C C V ( G ) , *6(B,C) is not

empty, then |» (B,C)| = Y D .

Proof. Let * € »Q(B,C), SCE(Gg), and define i)>' = <|>+i|/, where

39

( 1 , 1 ) i f e e S

( 0 , 0 ) i f eçES .

(23 )

(B,C) if and only if S is a cycle of

o-one correspondence "between

functions <j>' in 4> (B,C) and the cycles S of G , the lemma follows.

Thon i t is easily verified that <t>'*)

G . Since this establishes a one-to-one correspondence "between the0

For any graph G and any set ACV(G) let TI(A,G) denote the partition of

A induced by G, i.e. the partition in which two vertices of A are in tne

same "block if and only if they are in the same connected component of G. If

H is a spanning subgraph of 0, i.e. if V(H) = V(G), E ( K ) C E ( G ) , the

partition ir(A,H) is a refinement of it(A,G), i.e., the llocks of TT(A,H) are sub-

sets of those of TT(A,G). If H is a subgraph of G such that TT(A,H) = IT(A,G)

and no proper subgraph of H has this property, we call H a skeleton graph

associated with the partition TT(A,G). Evidently, a skeleton graph is a

forest, i.e., it contains no circuits. It is easily seen that for each

partition TT(A,G) with A#0 there is at least one skeleton graph.

The set of all partitions of A will be denoted by Í! , the set of all

even partitions of A (i.e. partitions of A into ever, subsets) by n , and

the set of all even partitions of A induced by spanning subgraphs of G by

n~(G). Obviously, = n„

Lemma 2 . I f , f o r S S O and f o r d i s j o i n t s e t s B , C C V ( G ) , 4> (E,!?) i s n o tD

er.pty, then ÏÏ(B,G ) and TT(C,G ) are even partitions.

Proof: Let H be any connected component of G„, BIT = BHV(H), and CTT =~~ 0 n Hcny(H). Consider a function $ 6 » (B,C). By eq. (21) we have

I Ot),vev(H)

I XV(B,C) = I (1,0) +vev(H) V S B „

(0,1) .

On the other hand, we have, by the definition of

*) If 3 is a cycle of Go, iji is a cycle vector of S, . If we denote theu y

number of independent cycle vectors (the cycle rank or cyclomatic number) of

a graph G by c(G), we have Ye = 2c ( ° e \ It is well known that c(G) =

|E(G)| - |V(G) I + number of connected components of G.

lt0

I O * ) v = I I *e = (0,0) ,

v £V(H) vEV(H) e inc v

since every edge in H is counted twice in the double sum. It follows that|B,f| and are even. Since H is arbitrary, the lemma follows.

Lemma 3. If, for 6 6 0 and for disjoint sets B, C Cy(G), 'M 2, 3 Uc) is not

empty and n(3, G.) is an even partition, then 4> (B,C) is not emcty.9 u

Proof. Suppose that *fi(0> BU C ) is not empty and that n(B, G ) is an even

partition.

Consider a skeleton graph H associated witli n(B, G„). Tlic- set E(H) is

the disjoint union of two uniquely determined sets E and S„: with the

property that if an edge e £ E (e^E.,) is deleted from H the connected

component of H containing e breaks up into two components, each one

containing an odd (even) number of vertices of B. Each vertex in B is

incident with an odd number of edges of E , each vertex in V(H)\3 is

incident with an even number of edges of £.. We now delete the edges of E 2

from H; in the resulting graph H' the vertices of B are the only vertices

of odd valency.

For v G B we have x (0, B ^ C ) = (0,1), xy(B,C) = (1,0) and hence

XV(B,C) = xv(0» B U C ) + (1,1).

Consider a function (0, B U C ) . We define a function

by

•j.' = $ + (1,1) if e £E ,

!)>' = <(> if e £ E ( G )\E1 .

Obviously $' satisfies eq. (15); by the above mentioned property of H' it

also satisfies eq. (21). Hence <)>'£ * (B,C), which proves the lemma. •

Let, for ACV(G) and d e n ° t e the set of all functions

0 S0 such that TT(A,G ) = ir and 4> (ø,A) is not empty,u U

Lemma 1*. For any A C v ( G ) , and any TVSII ( G ) , 0 (A) is not empty.

Proof. If n (G) is empty, the lemma is trivial, so let ^ « ( G ) bs non-

empty, and Tten^(G). Then, by the definition of II^(G), there is a

U1

spanning subgraph G' of G such that TI(A,G') = IT. Let H be a skeleton graph

associated with TT(A,G')> and E- and E o the corresponding edge sets defined

in the proof of Lemma 3.

We now define for each

0 if e?E U E O

1 if e£E ]

2 if e e E„

(25)

Z_ thus defined have

and for each e€E.|UE2

f (0,1) i f eSE

K =\_(0,0) i f e £ E 2 .

The functions 6 : E(G) •* {0,1,2} and <t> : E(GO) •* Z„o d

the following properties: (a) since G = H, we have TI(A,G ) = ir(A,H) =

TT(A,G') = -rr; (b) by its definition, $ satisfies eq. (15); (c) since the

vertices of A are the only vertices of G = H incident with an odd number ofö

edges of E1 (cf. the proof of Lemma 3), • satisfies eq. (21) with B = 0,

C = A. It follows from (b) and (c) that $£$A$,h), i.e. * (0,A) is notu o

empty, and from this fact together with (a) that 6 £@ (a), which proves

the lemma. •I t fol lows from Lemmas 1, 2 and 3 t h a t

I 14 . (0 , BUC) | i f n(B,G ) i s an even p a r t i t i o n

0 o t h e r w i s e .

From eqs . ( 12 ) , (22) and (26) we conclude t h a t

I » 6 ( B , C ) | =

(aT(BUC)

where for any set B CA and any partition IT 6 II

YDg(6)U (B),

(26)

(27)

A

r 1 if the number of elements of B in every block of IT is even

0 otherwise.

We observe that the factor in front of n (B) in eq. (27) depends only on

BUC, not on B and C separately. This implies t

introduced in eq. (6) can be written in the form

BUC, not on B and C separately. This implies that the function A.(G,K)

1*2

AA(G'K) = h I ? :

We now have the following two theorems:

& l'°-'B)(28)

Theorem 1. If A is an even set of vertices of a finite graph G, and

{X } c a set of complex numbers defined for all even sets BCA, with

A = *.\B for all B, then

BCA «•"*•>

for every Ising model on G if and only if

BCA V ' B

(29)

(30)

for every partition ÏÏ L (

) , then by eq. (28), AA(G,K)Proof. If eq. (30) holds for all TTG

vanishes (i.e., eq. (29) holds) for all K6 % Conversely, suppose that

A (G,K) vanishes for all K e 7i". Let IT be an arbitrary element of II^CG), H

a skeleton graph associated with IT, E and E_ the edge sets and 9 and $ the

associated functions defined in the proofs of Lemmas 3 and U. It was shown in

these proofs that 6 and <j> satisfy eqs. (15) and (21). On the other hand,

if $' is a function on E(H) satisfying eq. (21), the set of edges e with

•' í • must be a cycle of H. Since the only cycle of H is the empty cycle,

we have <j>' = <\>, i.e., $ is the only function with domain E(H) which

satisfies eq. (21). Eq. (15) then shows that, the function 9 defined by eq.

(2k) is the only element of 0 (A) such that 9 = 0 for eeE(G)\E(H), i.e.

if 9'€ G (A) and 6' 4 e then 9 ?! 0 for some eÊE(G)\E(H),

Let now K_ be a positive real constant and K the function on E(G)

defined by

Ke = KQ for e£E(H)

Ke = 0 for ee E(G)\E(H).

By eq. (10) we have in this case

(31)

(32)

where c = cosh K , sn = sinh Kn. On the other hand, we have y = 1. For"0 0

1*3

any other function 9'£fc) (A), g( 0' ) contains at least one factor c=s or s^

for e£E(G)\E(H), and hence vanishes. The same applies to any other

9'eo such that E ( G „ , ) \ E ( G J 4 0, in particular to any 9'eo ,(A} whereÜ V TT

TT' is not a refinement of IT.

Consider now a partition TT'SII (G) wnere T ' (5*11) is a refinement of TT ,

and a function 6'eø ,(A) such that E(0D, ) C E(G ). Fron the definition ofTI 0 u

the set E and the fact that TT' is even it follows that 0' is obtained from

6 by replacing the value 2 by the value 0 for one or more properly chosen

edges of E o. Hence,|Ej - |E |-r') = (coso) (s20) with r > 1 (33)

It follows that 2~ 2 | V ( G ) |(c 0s 0)-| E1 A (G,K) is a polynomial in s' of degree

in which the coefficient of the term of highest degree is

"BCASince A (G,K) = 0 for all values of the constant K , we must have

which completes the proof of the theorem.

Theorem 2. If A is an even set of vertices of a finite graph G, and

{X } e a set of real numbers defined for all even sets BCA, with Ag =

for all B, then

(31*)

for every ferromagnetic Ising model on G if and only if

(35)

for every partition n

Proof. For K E Tr we have _> 0 for all Therefore, if eq. (35)

holds for all Tien (G), then, by eq. (28), AA(G,K) is nonnegative (i.e.,

eq. (31!) holds) for all K £ / . Conversely, suppose A.(G,K) .> 0 for all

K E Í , Let TT be an arbitrary element of Jl ( G ) . Following the lines of the

proof of Theorem 1 we construct an interaction function E G ? for which

2 '(CQSQ) I I A A ( G , K ) is a polynomial in s . Since this polynomial

is assumed to be nonnegative for all nonnegative values of K„, the

coefficient of the term of highest degree must be positive, i.e.

f nn(B)A > 0 ,

BCA

which completes the proof of the theorem. g|

Corollary. If A is an even set of vertices of a finite graph G, and

{A } a set as defined in Theorem 2, then

BCA 3 B G A N B G

(36)

for every strictly ferromagnetic Ising model on G if and only if eq. (35)

(37)

holds for all nEII (0) and in addition

for at least, one i E B (C).

Proof. The corollary follows immediately from Theorems 1 and 2 together with

the fact that A (G,K) is an (entire) analytic function of all coupling

parameters. H

3. Generalization of Theorems 1 and 2.

Theorems 1 and 2 can be extended to identities and inequalities for

correlation functions on a vertex set A in which the products (a )(a ) refer

to subsets B and C of A which satisfy the condition that their symmetric

difference (to be denoted by BC) is a given set DCA; the case discussed

thus far, where B and C are disjoint and their union is A, corresponds to

the choice D=A.

It is possible to derive these generalizations by a proper extension of

the analysis of the preceeding section. However, for reasons of

transparency, and in order to show that the general case is, in a certain

sense, already included in the special case D=A, we shall present another

derivation, starting from the results of section 2.

Let A be a (not necessarily even) subset of a graph G, and D an even

subset of A. By n (D) we denote the set of all partitions of A in which the

1*5

number of vertices of D in each block is even, and by JI.(D,G) the set of

those partitions in HA(D) that are induced by spanning subgraphs of G.

Theorem 1 . If A is an arbitrary set of vertices of a finite graph G, D an

even subset of A, and {*RiRrA a- se^ °f complex numbers defined for all even

sets BCA, with A_ = A__ for all B, thensi JaJj

r \JoJJa ) =0 (38)BCA B B G B D G

for every Ising model on G if and only if

BCAnïï(B)xB = o (39)

f o r a l l p a r t i t i o n s i r e n ( D , G ) .

Proof. Let Q - A\D, and Q' a set of vertices not in V(G) which are in a

one-to-one correspondence with the vertices of Q; the vertex in Q'

corresponding to vGQ will be denoted by v'. Let G "be the graph defined by

V(G*) = V(G)UQ', E(G*) = E(G) U{{v, v'} | v GQ} , and let A* = A UQ'. We

extend the interaction function K to a function K on E(G ) by defining

if e e E(G)

> 0 if e£E(G*)\E(G) .

It is readily verified that for any set of complex numbers {A.},^. we have,13 ly-A

K* = Kg

K* =

I VVG*,K* (V\B )G*,K* = (1' cosh Ko sinh V I V V G . K ^ B D ^ . K '

(i*o)It follows that the sum in the right-hand side of eq. (1*0) vanishes if and

only if the left-hand side is zero, i.e., by Theorem 1, if and only ife * e / *

Ijjj-.T) (B)}- = 0 for every IT EJ t(G ); strictly speaking this requires the

symmetrization of the set i gl-or« with respect to A .

Consider a partition IT £ Hj((C ). Since TT is an even partition no

vertex v'e Q' can form a block by itself. Therefore every vertex v'€ Q' is

in the same block as the corresponding vertex v£ Q; hence, the number of

elements of D in every block is even. Let now IT be the partition of A

obtained from IT by deleting all vertices of Q'. Obviously, irEII (D,G).

Conversely, every TT€II.(D,G) can be supplemented to a partition IT 6 L „ ( G )

by putting each vertex v'G Q' into the same block as the corresponding

vertex vGQ, and hence the theorem follows. •1*6

The corresponding generalization of Theorem 2 (to be referred to as

Theorem 2 ) is obvious and will not be discussed explicitly. Eq. (38) and

the analogous generalization of eq. (3M will again be called a A-identity

and a A-inequality» respectively.

Theorems 1 and 2 remain valid if the unnormalized correlation functions

are replaced by the corresponding normalized correlation functions, provided

we restrict ourselves to Ising models with Z^O; in Theorem 2 this

condition is always satisfied. The resulting identities and inequalities

remain valid if for some edge e = {u,v} we take the limit K i.e. if

the edge e is contracted and the spin variables a and o are identified.u v

Using eqs.. (28) and (hi) and the relations

3g (D= 1 + 2g (2),

e

J2)

3K 3Ke e

we also find that taking the derivative with respect to any coupling

parameter K (eEE(G)) in a A-identity (A-inequality) results in a A-identity

(A-inequality). For A-inequalities this implies, in the terminology of

Newman5)

that they apply strongly.

h. Some properties of sets of A-identities.

It follows from Theorem 1 that for a given graph G and a given set

the number of linearly independent A-identities, with D=A, for

spin correlation functions on A, to be denoted by L (G), is equal to the

number of linearly independent solutions of the set of linear equations (30).

The latter number depends on G and A only through the set of partitions

H (G). In general, the larger the set n (G), and hence the number of

conditions on A, the smaller the number of linearly independent solutions of

eq. (30), and hence L ( G ) . In particular, we have the following theorem.

Theorem 3- If G and G' are f in i te graphs, and A i s an even subset of V(G)

and V(G'), then

(i) L (G) = 0 i f n^(G) = ne. (Ma)

(ii) < LA(G) i f

i i i ) LA(G) = 2 |A|-2 .i f n„(G) =

(Mb)

(Me)

Proof. (i) If n (G)=n , eq. (30) is required to hold for all even partitions

ii of A. We shall show that already the set of equations obtained by

restricting TT to the partitions of A into one or two even subsets has no

nontrivial solution.

If P and A\P are the subsets into which A is partitioned by TT(where forTT TT

convenience a one-block partition is considered as a partition of A into 0

and A ) , then for any even set B C A

'i if |ßnp I is evenn (B) =

0 if |Bnp | is odd ,

i.e.,

njB) = id + (-1)IB"^I) .

The set of eqs. (30), with the restriction imposed on TT, can therefore be

written as

= 0 (1*2)

BCA

for all even P C A ; observe that the equations with P and A\P are identical.

Multiplying eq. (1*2) with (-1) , with B ' C A , |B'| even, summing over

P and using the relation

PCA

for U=B' and U=BB', we obtain

BCA ',A'

It follows that A , + A y , = 0 for all even sets B' CA. Since we have

taken A , = X . ,, it follows that A , = 0 for all even sets B'CA. Hence,B' AA\B B

LA(G) = 0.

(ii) If n^(G) cn^(G'), the set of linear eqs. (30) for the graph G is a

(proper or improper) subset of that of G', and hence the set of identities

of the type (29) valid in G' is a subset of the set of identities valid in

G. Statement (ii) follows.

(iii) If "„(G) = 0 there are no linear equations for the coefficients

(except the condition A_ = l&r, for all B ) , and we have LA(G) =

I A I A§|{B|BCA,|B| even }| = 2 | A | " 2 . The simplest set of independent identities

then consists of the equations 'CTB^G^°A\B^G = ° f o r a 1 1 K G ^

1*8

(RCA, E even), the validity of which is trivial.

Examples of the three cases considered in Theorem 3 are:

(i) G is a complete graph, i.e. G= (V,E = ) ,'*itn E o = {{v,v'}| v, v'GV, v£v''.

(ii) G is a spanning subgraph of G'

(iii) G is an empty graph, i.e. G = (V,0).

If G = (V,E) with E ? E c there is at least one set A C V for which

L (G) >0, viz. A=V. This is expressed in the following lemma.

Lemma ?• If G = (V,E) with E = E c \ e (e 6 Ec) we have the following identity

BCV " u " " J

here v. and vo are the vertices incident with e.

Proof. The equations (30) read in this case

= 0

BCV

r n ((V 1,V_}UR) - 1°RCV" * 1 2 RCV"

= o (MO

where V" = V\{v ,v },and ZT1 d n,H denotes summation over all odd subsets of V".V

Consider first a partition TT in which v and v 9 are in the same block.

By the structure of G, partitions in which v and v„ form a block by them-

selves are not contained in nfl(G). Therefore, if we denote the blocks of IT by

U.|,U„,...,U (where U. is the block containing v. and v ), U. n V" is not

empty. The first sum on the left-hand side of (MO is equal to the number of

even sets R C V " such that |R nU.| is even for 1<_i<^r, the second sum is

equal to the number of odd sets R C V " such that I R ^ U . I is even for 2 < i < ri — —

and odd for i=1. Since U ny" is not empty, the number of even subsets of

U ^V" equals the number of odd subsets. Hence, the two sums in the left-

cancel. A similar argument applies to partitions in whichhand side of

and v 2 are in different blocks.

The generalization of Theorem 3 to the case that A is an arbitrary

vertex set of G and D an even subset of A is straightforward.

h9

/

Examples of A-identities.

1) As a first example we discuss in detail the case |A| = h, D = A. Let

A = tv1,v2,v,,v)t} . We have, with A_.. =1

v,,v) _ { — } and

X13(a1a3)(a2al*)

The set n.(G) is a subset of the set H = {TT . 10 <_ i <_ 3} where, in an obviousA A i

notation,

), *, = (12|3M, ^ = '' 3= (IUI23). If we define:

(1*6)

we find,

13

a) If H.(G) = n", then according to Theorem 3 there are no A-identities for

A, as can easily be checked by putting all k (ir.)> Of.ii.3, equal to

zero. As remarked above, this case is realized e.g. if G is a complete graph,

with A CV(G).

b) We now consider the cases where |n (G)| = 3. One easily verifies that

the only possibilities are K?(G) = n^\{ir.} with If.i.f.3» since TT is in

n (O) whenever two of the three partitions TT. ( 1 _<_ i _<_ 3) are in n (G). Without

lack of generality we assume RÀ

existence of an identity now consists of the equations k (IT.) = 0 (i=0,1,3),

which have as the only solution

identity reads

T"3

A

} . The condition for the

The resulting13

gCJ^) - (a1a2)(a3alt) + (a = 0 .

The graphs G for which this identity holda are characterized by the fact

that the partition TT of the set A is not induced by any spanning subgraph of

G. This implies that every chain between v and v, separates v from v.

•50

(i.e. has a vertex in comr.icn v.ith every chain between Vp and v, ).

The identity ('J8) is a special case of a general class of identities

which formed the subject of I, and to vhich we shall return later on in

this section (example 1*).

c) Consider now the cases where |J*I~(G)| = 2. The only possitility is that

where IT. (G) = {ÏÏ ,ÏÏ.} for some i (1 <i< 3). Suppose i=1. From theA u 1 — —

equations k.(ir0) = kA^"i' = ° w e fi Rd ^ = ~ Ai2' 'S 3 = ~^1a' a n d ' n e n c e.(ir0)

- (cr^Ma a ] + X

for any L and X.,- This implies

3)] = o

(1*9)

(50)

This case applies when G contains a vertex v (cut vertex or articulation

vertex) which separates v. and v from v and v, (i.e., which is contained

in every chain between v and v etc.); v need not be distinct from v ,v„,

v, or Vi. This can be shown by introducing two new vertices u and u', and

four new edges {u,v.}, {u,Vp}, {u',v^}, {u',v, } and applying Menger's

theorem to the vertices u and u' (cf. ref. 2, p. 129).

The relations (50) are trivial in that they follow directly from the

factorization of correlation functions in a graph with a cut vertex,

d) If |n^(G)[ = 1 we have either 1^(G) = Í^Q) > °r n ^ G ^ = { l Ii } w i t h

i=1,2 or 3; suppose i=1. In both cases there are three linearly independent

A-identities. In the former case we find

this case applies when G contains a cut vertex v (not necessarily distinct

from v.,v ,v or v. ) which separates every v. from every v. (1<i,j<U,i^j).1^34 1 j — —

(51)

If IT is the only partition in H^(G), we find

= (a1o2)(a3alt)

(52)= 0

are in one component and v, and v, inHere, G is not connected, v and

another one.

e) The case where n A o ) is empty, has been dealt with in Theorem 3.

51

2) V.'-i r.ext. consider the case A = {v ,v2,v jV^}, |D| = 2, e.g. D = {v ,v9}.

The set ^(D.G) is a subset of the set {(1231*), Í 1231 U) , ( 1 £-U | 3) , ( T2 | 3^),

(11-|3|;()}. Again, if n (D,G) contains all these partitions, there are no

A-ider.tities. In contrast v.rith example 1, the deletion of or.e partition

from this maximum set does not necessarily lead to a A-ider.tity. If, e.g. >

the partition (121»|3) is missing (which is the case if v_ separates v ar.d

v, from ' / , ) , one easily sees that the eqs. (39) have ne non-trivial

solution.

V.'e discuss only one case- in which idciititi'.G do cccur, viz. that where

v, separates v from v, and v^. In that case, n (D,0) = {(K3'O, (123|Mh

'i'i.i-'rc are two conditions on the four independent coefficients X , and we

find the following identities,

(53)

w:iich also follow, of course, from the factorization property mentioned

above.

3) We new turn to the case |A| = 6, D=A. If A = {v. | 1 ±i ±6}, then

AA(O,K) = 2{ó

V.'e give one example of a A-identity which can occur in this case, viz, that

with

= A

X 1 2 =

13

= A15 = A16 = A 36

The corresponding numbers are non-zero only for the partitions

= (I3|?ii56), TI.D = (1312't j ?6), = {13! 25 | i+6) and IT = (13[2611+5) - Hence,

the corresponding A-identity holds if these partitions are not in II (G).

This car. occur in various ways, of which we mention the following ones.

a) G is the graph obtained from the complete graph Kg by deleting the edge

( v ^ v , ) , and A = V(G). This case has been dealt with in Lemma 5.

b) G is a planar graph and the sequence (v.,vo,v ,v.,v ,v>) is a boundary

sequence of 0. A boundary sequence of a planar graph G is a sequence of

(not necessarily distinct) vertices (u^-.-.u ) of G such that the graph G1

defined cy V{';') = V(G)

" i '

. . .w^} (v. f V ( G ) , 0<_i<_r.) and U(G') =

. )Í> <i<r.} \.-ith w ..l + I ' n.+ I

(s-=e I ) , i t was proven in I t h a t for any boundary sequence (u.. , u ; , u , , u . )

every chain between u and u., s epa ra t e s u3 from u. ; fur thermore , every

subsequence of a boundary seouer.ee i s a boundary sequence. Applying these

p r o p e r t i e s t o the sequences (v , v , , v , v . ) (i = 'i,'},(>) we .'C- tha t indeed 'a'.i'ji

do-'C not cor.tain the p a r t i t i o n s i\r , i;_ , n and TT .

This example car. be g e n e r a l i s e d , e . g . in the fol lowing way. Let

(v-, j v ,, v , v, ,v:.', v ,v- ,w, ' ) be a boundary sejuence of a p lana r graph G , G an

a r b i t r a r y graph such t h a t V(G1 ) n V('J ,) = fvr ,V(.,w',•••'} , and G = j U 0. ,

t he union of G and G,,. A chain between v and v which does not contain

t:j; v e r t i c e s v ,Vi ,vr_,v i s e l f n e r a chain in >} or i t c-nta!r.3 •-•d^os ^f " ,

and iier.ee •.•:' and w' . In the former case i t s e p a r a t e s v l'rom v, ,v. and v, ;

in the l a t t e r case i t con ta ins a chain between w' and v , which s e p a r a t e s

v, froiu v^, v( and v - . In botii cases ÍI (G) does not contain i he p a r t i t i o n s

"a' V ïïc ^na "d'

h) As a final example of A-identities we now give ar. alternative proof

of the main result of I mentioned in the introduction. Let 3 be a finite

planar graph and (v.,...,v ) a boundary sequence of G. 3y Theorem 1 , the

A-ider.tity

J = 1

holds if and only if

o )(o a n a ) = 01 j 1 j k=1 k

(55)

(56)

for all partitions TT£JI,(D,G), with A the set of aJl vertices occurring in

the boundary sequence and D the set of vertices occurring ar. odd number of

times. Let Tt be such a partition and let v. , v. , ..., v.

. J1 . J2 , "m( I < j 1 < . . . < j <n, m odd) be the v e r t i c e s which are in the same block of

I m ~~it as v . Obviously, us ing the fac t t h a t n (0) = 1, eq. (5&) reduces t o'1

m

r=0(57)

where j =1. Since for any r every chain between v. and v. separates allJr Jr+1

v. with j < i < j from all other vertices of the boundary sequence, the

53

set {v.|j <i <j ,} must be a union of blocks of TT, and hence even. There-

fore, (-1) r = (-i)r~ , which proves the validity of eq. (56), and hence

of eq. (55).

6. Examples of A-inequalities.

1) First consider the first GKS inequality restricted to ferro-

magnetic Ising models on a graph, i.e. with pair interactions, which we

write in the form

(°0^<V - ° ' (58)

Since n (0) = 1 for all TT e n^, eq. (58) follows from Theorem 2.TT A

2) The second GKS inequality for ferromagnetic Ising models reads

'V'Vc' " ''W - ° ^59^for arbitrary sets B, CCV(G). The validity of this inequality in the case

of pair interactions follows from the fact that ti (0) = 1 and ri (B) = 0 or 1,

and hence n (0) - \^) 2. °> f o r a 1 1 ïï 6 "BUG (BC>G).

3) Vie next turn to a set of inequalities recently derived by Newman for

ferromagnetic Ising models with pair interactions '. Let A be an even

subset of a graph G and X^ a collection of even subsets of A such that every

partition of A into pairs is a refinement of some two-block partition

(B|A\B) of A with B£X,. Newman's inequality reads

B£X,

By Theorem 2, it is sufficient to prove that

(60)

(61)n Í 0) = 1 < F n (B) for all iren»(G) .TT '— TI A

A

Consider any IT e II. (G) and let TT' be a partition of A into pairs which is a

refinement of u. Since, by definition, there is at least one B 6 X such

that TT' is a refinement of (B[A\B) every block of TT' contains 2 or 0 elements

of B. Hence n ,(B) = 1, and therefore n^(3) = 1. Sines n (B')^O for all

other B'6>; the inequality (59) holds.

Our general formalism, applied to this particular case, resembles the

derivation of eq. (6o) given "by Sylvester

h) Finally we derive a new A-inequality for the case |AJ = 6 , D = A

n ame ly:

6 6

A • „ 1 j 1 j A — . . „ i j i j A

By Theorem ? It is sufficient to show that for all u E J

(62)

(63)

If i = (1231456), v.'e have n ({v.,v.}) = 1 for all i,j (if j) and hence eq.(63)

is valid. If IT is of the type ( 1 j | kJlmn), the left-hand side of eq. (63) is

equal to 2, the right-hand side to 6, and again eq. (63) is valid. The

remaining two cases, where TT is of the type (1jki|mn) or ( 1 j|jc£ |mn), are

dealt with in a similar way. The extension of the inequality (62) to the

case |A| = n>_6, n even, is straightforward.

The examples of A-inequalities discussed above are valid for any graph G

containing the given sets A and D. Evidently for any choice of A and D,

Theorem 2 enables one in principle to derive all A-inequalities on A which

have the same general validity. The corresponding sets X, considered as

vectors in R (n=2' c ) with components X , form a convex cone. If, for a

particular choice of G, II (D,G) is a proper subset of n (D), the

corresponding set of A-inequalitites will again form a convex cone, which

may contain the convex cone mentioned above as a proper subset.

In a subsequent paper we shall investigate the structure of these

convex cones. In particular we shall show that every A-inequality can be

"decomposed" into a finite number of extremal inequalities. We shall discuss

the relation of the inequalities mentioned in the examples with these

extremal inequalities, and derive new (extremal) inequalities.

55

7. Ccr.'-'ludi.iH remarks.

1) Formally, the analysis of this paper is restricted to Ising models in

zero magnetic field. The case of an arbitrary (not necessarily homogeneous)

magnetic field can, however, be easily included by replacing the field by a

"dummy" spin interacting with all other vertices

.-) Theorems 1(1 ), 1{('<1 ) and 3 have beer, derived for finite graphs. The

extension to infinite graphs is straightforward. Consider an infinite graph

G, two finite sets cf vertices ACV(G) and DC A, and for each, partition

TIE.", (G) a skeleton graph H_ . The edge sets of all H SSK- finite. Hence

there exists a finite subgraph 0 of G sucli that E(:i ) J0' for all n. Let

0 , G , "i j5 ... be a sequence of increasing subgraphs of G such that

lira 'j = G and that (o > = lim <o > exists for ail BCA. Thenn-x» n B G n-«° B Gn

I1*T(G ) = Jl (G) for n = 0,1,2,..., and the validity of the theorems (with the

unnormalized correlation functions replaced by normalized correlation

functions) for infinite graphs follows.

3) As the reader may have observed, all examples of A-identities given

in section 5 have the property that all A are equal to 1, -1 or 0. In a9) . .

forthcoming paper , devoted to A-identities for Ising models with general

(n-spin) interactions, it will be shown that for every choice of G, A and D

the set of all A-identities can be derived from an independent set of

A-identities having this property.

h) !-'any theorems on inequalities include a specification of the condition

under which the equality sign holds (cf., e.g.,ref. 10). For correlation-

function inequalities such conditions have hitherto not received mach

attention in the literature (see, however, ref. 11). In the case of

A-inequalities they are implicit in Theorem 1 , since the validity of a

A-equality for all K G / implies the validity for all KG-a". The condition

GKS inequality,

e.g., the condition reads (cf. section 6, example 2):n (B) = 1 for all

TT e n_. „ (BC, G ) , i.e. all partitions of B UC for which n (B) = 0 are absentKl * TT

C = {v,,Vi }, the missing

partitions are (1312U) and (1U j 23)s in which case G contains a cut vertex

separating v- and v from v and v. (see section 5, example 1c). In case G

contains a "dummy" vertex v connected with all other vertices, representing

a nonzero magnetic field, v must be the cut vertex. The graph obtained

from G by deleting vQ and the edges incident with it then has v and v in

takes the form of a condition on n (D,G). For the second

from HBUr(BC, G). E.g., if B =

one component,

derived by Setô

v, and in another. This specific result vas earlier

Heforences.

I) J. Croeneveld, R.J. Boel and P.W. Kasteleyn, to be published in i-hysica.

. ) R.J. Wilson, Introduction to Graph Theory (Oliver tBoyd,Edinburgh, 1'.'72 ).

3) F. Harary, Graph Theory (Addison-Wesley, Reading, l-'ass., ï'ji'O) Ç- 37 ff.

h) R.E. Griffiths, in: Phase Transi 'ons and Critical Phenomena, Vol. 1,C. üomb and M.S. Green eds. (Academic Press, London, 1972) p. 7" ff.

'.<) C. Ile'-nnan, Z. Wahrscheinlichkeitstheorie verv.'. Gebiete J33 (1975) Vj.

ij) J.J. Sylvester, Comm.Math.Phys. hZ_ (1975) 2Ü9.

7) ?.W. iCasteleyn and R.J. Boel, to be published.

3) P.W. Kasteleyn, in : Graph Theory and Theoretical Physics, F. Karary ed.(Academic Press, London, I967) p. U3.R.3. Griffiths, J.;-!ath.Phys. B ( 1?67) h8k.

9) R.J. Boel, to be published.

13) ï.F. Beckenbach and R. BellEian, Inequalities (Springer-Verlag, Berlin,' 1361).

II) U. Seto, Progr.Theor.Phys. .55.(1976) 683.

57

III. EXTREMAL A-INEQUALITIES FOR ISING MODELS WITH PAIR INTERACTIONS

1. Introduction.

In this chapter we shall investigate in more detail the class of

correlation-function inequalities introduced in the previous chapter. In

particular, we shall show that every inequality in this class which refers

to a particular set of vertices A can be written as a positive linear

combination of so-called extremal inequalities with respect to the same set

A. A method will "be sketched by which they can, at least in principle, "be

found. Throughout the chapter we restrict ourselves to the case discussed

in Theorem 2 and section 6 of chapter II (i.e. D=A). The generalization to

the more general case (i.e. DCA) can in principle be treated in a similar

way and will not be discussed here. Some explicit examples of extremal

correlation-function identities for general A will be given; for the case

|A)=6 we shall in addition give three examples not covered by the former

ones. These extremal inequalities are the strongest correlation-function

inequalities in the subclass determined by A.

2. Definitions and notation.

As in chapter II, a graph G is defined as a pair ( V ( G ) , Etc)), where

V ( G ) is a set of elements called vertices and E(G) a set of unordered pairs

{v,v'} of distinct vertices, called edges. G is finite if V(G) and E(G)

are finite.

An Ising model on a finite graph is defined as a triple (G,/,K), where

•/is the set of all functions a: V(G) •*• {-1,1} and K a complex function on

Etc). The spin variable a is the value of o at the vertex v, the coupling

parameter K is the value of K at the edge e. An Ising model is called

ferromagnetic if Kg>_0 for all e£E(G).

For any set ACV(G) we define

a, = IT aA v£A V

(1)

58

where for A=0 we have 0^=1.

The Hamiltonian of the Ising model (G,/,K) is defined by

Ï K a

t h e u n n o r m a l i z e d and n o r m a l i z e d ( s p i n ) c o r r e l a t i o n f u n c t i o n s ( a . ) _ „ andA li,K.

<a.>r ., , respectively, for any set ACV(G) by-H„ „(o)

( 2 )

(3)

'V« Z " '

where Z, the canonical partition function, is defined by Z = (1)„ „• Since

the Hamiltonian is quadratic in the o , the correlation function (o.)V A u,K.

vanishes if |A| is odd. Therefore, we shall henceforth consider only

correlation functions {o.)r for even sets A, i.e. for sets with |A| even.

3y ? ( A ) we denote the family of even subsets of A. We shall suppress the

index K and, where no confusion arises, the index G as well. We have taken

kT=1.

Consider a graph G and a set AcV(G). By ?r(A,G) we denote the partition

of A induced by G, i.e. the partition in which two vertices of A are in the

same block if and only if they are in the same connected component of G. If

H is a spanning subgraph of G, i.e. if V(H)=V(G), E ( H ) C E ( G ) , the

partition TT(A,H) is a refinement of TT(A,G), i.e., the blocks of TT(A,H) are

subsets of those of TT(A,G). The set of all even partitions of A (i.e.

partitions of A into even subsets) is denoted by n , and the set of all even

partitions of A induced by spanning subgraphs of G by n (G). Vie furthermore

introduce for any set BCA and any partition nSH.

1 if the number of elements of B in every block of TT is evenn (B) = -i

1 0 otherwise.

Observe that n (A\B) = n (B). In II we have derived the following theorem.

Theorem 1. If A is an even set of vertices of a finite graph G and

set of real numbers defined for all sets Be,p(A) with *B=*A\B

f o r a11

then

(U)

59

for every ferromagnetic Ising model on G if and only if

[ n (B)A >. 0" B

(5)

for every partition irEIl (G). The equality sign in eq. (It) holds for every

Ising model on G if and only if the equality sign in eq. ( ) holds for every

partition TTSTI . (G).

Every inequality of the form eq. {k) wil_ be called a A-inequality. The

set of eqs. (5) is a finite set of linear inequalities. In the following

section we shall present some general properties of such sets of linear

inequalities.

3. Polyhedral convex cones.

Consider for any vector xSflin the following set of linear combinations of

the components x. of xs where I is a finite index set.

I a. x ; iel ,k=1

(6)

with ct.,E|R for all i and k. We define the setIk

n

Ik=1

.> 0 , for all ielj . (7)

The set C is a convex cone in the sense that if x ,x £C then every<* (•,) (2) a

positive linear combination of x and x (i.e. every vector(1) (2)

e x + CpX with c ,Cp^_O) also belongs to C . In view of its definition

and the fact that I is finite C is called a polyhedral convex cone.

For any JCI we define

= {xe&,n | I cu x > 0, ieJ ;k=1

a x = 0 , for allík "k

. (8)

If FT is nonempty it is called a face of C ; Frf is called the null-face.

If d. is the number of independent vectors satisfying E a. x_ = 0, for all

i£l\J, we call FT a face of dimension dT. It is clear that there are noJ J

faces of dimension smaller then d=dj, that there is exactly one face of

dimension d, and that there are at most 2' ' faces.

60

A face FT of C is called an extremal face of C if no vector in FT canJ a a J

be expressed as a positive linear combination of two vectors in C \F T.

'tis now state without proof the following facts about C :

1) The extremal faces of C are the d-dimensional null-face F„, and theex 50

{d+1)-dimensional faces (if any). Vectors in (d+1)-dimensional faces we

call extremal vectors.

O If in each extremal face F with d.=d+1 we select an arbitrary vectorJ ( 1 ) ( d ):< , and if the vectors x ,...,x form a basis of F^, every vector x inC jan be written in the forma

.. _ y c xJ + y c x ( r ) (?)X - I L A I" I C X , \j)

JCI:dr=d+1 r=1

with cT >0 for all J. For d > 0 this decomposition of x in terns of extremal

vectors and vectors in the null-face is not unique. For d=0 the r.ull-face

consists only of the null-vector, and the decomposition is unique (apart

from a positive normalization factor in each of the x ; in this case, to

which we shall restrict ourselves in this paper, the x can be formed by

the following procedure. Select in all possible ways a set of n-1 linearlyn

independent equations Z,_.a.,x = 0 ; if a vector x satisfying these

equations lies in C - it is an extremal vector. For a general referencea

to linear inequalities see ref. 1.

Extremal A-inequalities.

In this chapter we shall consider graphs G and sets A for which

3) = n ,

inequalities

n (3) = n , and in this section we sha." 1 investigate the set of linear

Ie," ( A ) 3

iren. do)

Because n (3) = n (ANE) and X_ = A . for all Be'? (A), we select from theIT TT n A \D e

even subsets of A a maximal family of subsets J"e(A), such that if Be.?'(A)

then A\B£V'(A); we assume that 0e^(A).

Let us denote the two-block partitions of A by {B,A\B}, B € J " ( A ) . If for

convenience we write the one-block partition {A} as {0,A} we can introduce

*) This proviso will not be repeated explicitly in the sequel.

61

a square matrix n with elements

From the definition of n (B1) it follows thatTT

(-1) ) ; (12)

-1It is not difficult to see that n has an inverse n with elements

, = -Si 13,0

UB',{!

Let us introduce the set X = {XD iBe?' (A) }, considered as a vector in

. ,„,. - r. ). The set o

L e t us i n t r o d u c e t h e s e t X = {Xß | B e j (A) }, c o n s i d e r e d a s a v e c t o r i n

lf t n (n = | ? ' ( A ) | = 2'A<~2). The s e t o f v e c t o r s X w h i c h s a t i s f y e q . ( 1 0 ) f o r

a l l will form a convex cone C . It is convenient to carry over then

concepts "extremal" and "positive linear combination" from the vectors A to

the corresponding A-inequalities.

We can now combine the results of the preceding section with Theorem 1.

Since ri has an inverse, we are in the case d=0, and the following theorem

holds.

Theorem 2. Let G be a finite graph and AcV(G). If n![(G) = n*T, then every

A-inequality with respect to A is the positive linear combination of a

finite number of extremal A-inequalities with respect to A.

To illustrate Theorem 2, we consider the case of a set A consisting of

four vertices, say A = {1,2,3,1*}. n, consists of the following partitions

(in an obvious notation)

| 12310, , ( 1 3 | 2 U ) , (11*)

Let j " (A) = {0, {1,2}, {1,3}, {1,1*}}. The matrix n has the following form

(15)

where the order of rows and columns is that of the partitions in {ík). The

extreme vectors of C can be found by selecting any three linearly

independent equations from the set of four equations

62

I r, A = O , Be?'(A)?;(A) B'B 3

and requiring that the solution satisfies

^ B B B

for the remaining sets B. In this way we easily find the following extremal

vectors (normalized so that |X-,|=1):1 in

~ ~X12 = A13

= ~X12 = "X1

X0 = X1

X0 = "X

(16)

3'

3>

3)

3>

1 0 ,

1 0,

1 0,

i 0.

(

(

(

(

17a)

17b)

17c)

17d)

The corresponding extremal A-inequalities are

( O f o ^ a ^ ) - (a1ti2)(o3alt) + (o^^io^e^

gO ø^) - (criff2)(o a^) - (c^a Koyj^

-(•\)(o^o2o3ah) + (cr^Ka^) + (a^a^ia^^) + (a^^J

The inequalities (17a, b, c) are special cases of inequalities aerived by2)

Sherman (see also ref. 3 and ref. h, Theorem 5). The inequality (17d) ish)

a special case of a set of inequalities due to Newman , and is also a

consequence of the stronger GHS inequality (see also refs. 6 and 7). By

adding all four eqs. (17) we obtain

(D(a1<J2a3a1() >. 0 ;

by adding eqs. (17a) and (17b) we obtain

(18)

(19)

Inequalities (18) and (19) are examples of the first and second GKS

inequality , respectively, which are thus seen not to be extremal. We

stress the fact that for the given set A the set of inequalities (17)

exhausts the class of extremal (i.e. strongest possible) correlation-

function inequalities of the form (M. The example of the GHS inequality

shows that there exist stronger inequalities which are not in this class.

To prepare the way for the examples to be discussed in the next section

63

ve find it convenient to introduce a change of variables which enables us to

write the inequalities in a somewhat simpler form. We define, for BeJ"(k),

B B'GÍ;(A) B > B B

Using the inverse of n it is easily verified that

XB = -6

B'ej"(A)

(20)

(21)

Consider a partition ir = {B ,. . . ,B, } e H^, with B ,...,& nonempty. Fron

the definition of r\ it follows that for BeJ" (A)

nw(B) = n (id + ;-D m )\ .

Define K = { 1 , . . . , k } , and for LCK def ine

B L = U B £ '

(22)

(23)

( i n p a r t i c u l a r , Brt = 0 ) ;v

n (B) can now be w r i t t e n as

- k ^ ln ( B ) = 2 K I ( - 1 )

LCK

Using eqs . (21) and (2M and the fac t t h a ta }

X ' A(25)

for a l l X e j i (A) , we f ind

BeiV n (B)AO = - 2 "W 1 V B

(-1)LCK

BGJ^(A) B?G?^(A) LCK

+ 2-k+2 k (26)

If we introduce

(B) =j 1 i f B Is a nonempty union of (nonempty) blocks of TI

(_0 otherwise

•:e finally obtain

(27)B6>;(

In the new variables the set of inequalities (10), with j> (A) replaced uy

,?'(A), reads

Bei'(A)(B)k (28)

Note that the set of inequalities (28) includes the set of inequalities

k > 0, for all Be?'(A).

To find the extremal inequalities from (28), we select a set of n-1

independent equations of the form

„e4, !. ( B ) k- = ( !• <

(29)

If the solution of eqs. (29) satisfies the inequalities (28) for alle .

remaining ir ir. II , then it will be an extremal inequality.

5. Examples of extremal A-inequalities.

In this section we shall give several examples of extremal A-

inequalities. The examples given do not exhaust the class of all possible

extremal A-inequalities.

1) We shall first derive three types of extremal A-inequalities which are

valid for arbitrary sets A with |A| >_k. To this end we first select a

vertex v£A and choose for j"(A) the set of all even subsets of A note \P -°

containing v. As in section '* we define n = |-i"(A)| = 2' ^.

(a) Let C be an arbitrary nonempty element of J"(A), and take

k^ = 1 for B=C ,

k„ = 0 for

(30a)

(30b)

Obviously, since 0, all inequalities (28) are satisfied. Using eq.

65

(2i)ve see that the A-inequality corresponding to the choice (30), multipliedIAI — 3

by a factor 2' ' in order to avoid fractional coefficients, reads

l ("D (31)

Since the n-1 eqs. (30b) are manifestly linearly independent, the

inequality (31) is an extremal A-inequality. Any A-inequality with k^=0 and

k #0 for more than one Bep'(A) is a positive linear combination of A-

inequalities of the type (31) with strictly positive coefficients, and

hence not extremal.

It is easy to verify that the inequality (31) is of the type derived by2)

Sherman (see also ch. IV).

(b) Next we select a vertex v'eA, v'^v, and we take

k„ = 1 if v'Ö and B 4 A\{v,v'} , (32a)a

k = 0 otherwise. (32b)

Obviously, this set of k_-values satisfies the inequalities (28) for all TT

such that k = U | £2. If |A| I_G we further consider a partition

TI = {B ,...,B } with k > 2 , where the order of the blocks is chosen so that

v,v'eB UB . There are two cases to be distinguished: (a) v and v' are in

the same block, say B , and (ß) v and v' are in different blocks, say v in

B., and v' in B^. In case (a), the left-hand side of eq. (28) is equal to

2 k - 1 - 1 (if B1 4 {v.v1}) or equal to 2 k - 1 - 2 (if Bj = {v.v1}). Since k. = 1

and k > 2 , the inequality (28) is satisfied. In case (ß), the left-handk—1 k—2 k—2

side is equal to (2 - i ) - 2 = 2 - i , and hence the inequality (28)

is again satisfied.

To show that the choice (32) of k_-values satisfies a set of n-1n

linearly independent equations of the type (29) we consider, for |A| f^6, the

partitions TT = {B ,B ,B,}, with veB^ v'eB„. For such a partition the

equation (29) reads

(33)

which reduces, by eq. (32b), to the equation k =k_:- Since for any

B 4 A\{v,v'} not containing v' such a partition with B =B can be found,

and since the full set of equations for the k_ thus obtained is linearlyB

independent, and has (32) as a solution, the corresponding A-inequality is

extremal. By using eq. (21) one easily verifies that the coefficients X

(again multiplied by a factor 2

are

f 2 IA | - 3 -

X = j 1

in order to avoid fractional values)

if B = A\{v,v'}

if B contains v'

otherwise.

(c) Finally we take

kB = o if |B| >J - 1 (35a)

(35b)

(35c)

k B = 1 if |B| = a I A I -1 ,

kB = 2 if |B| <J|A| -1 .

Again, this set of k -values satisfies the inequalities (26) for k= |TI

If IA I > 6 we further consider a partition TT = {B ,. . . ,B } with k > 2 and

veB.., and we define M = {2,...,k}. Using the definition (23) we write eq.

(28) for this partition as

M LCM

Since for any L

'B.,) > !36)

BM\L

,M we have

B.'M\L'

= |A| - |B,| ± |A| - 2,

|B | and | 3M w | cannot be both larger than J|A|-1. Hence we have, by (35),

k„ + K, * 2. Since the number of sets L 0.M is 2' ' - 2 = 2 ~ - 2 , andBL % \ L -since k =2, the inequality (36) is satisfied.

We shall now show that the set of k defined by (35) satisfies a set of

equations of the form (29) with k = |ir ^_3, among which n-1 are linearly

independent. First, the eqs. (35a) are of this form, and they are

independent. If |A| >_6, we further consider the set of simultaneous

equations of the form (29) where IT = {B,B,B,} , with B- = {v,v'}, where

v' is an arbitrary vertex not equal to v. They have the form

R U RB2 U B3

B2

which reduces to

\ + \ = k (37)

since |B„ÜB,| = I Al - 2 > 5 Ul -1. If I B„ I > IB O| , then kD = 0 , and eq.(- J i- J> -Dp

(37) further reduces to the equation k =krt. If B |B I we consider a

67

vertex v"s3 ar.d the sets B' and B' obtained from 3 and B by interchanging

aud v

K, +

For the partition ^,B^, B } eq. (:>) reads

!38)

From eqs. (37) and (38) it follows that k , = kT . Repeating this argumentBp iio

we find that k_, = k„ for any two sets B,B' .-.'it;."| B _, | = |B | considered we conclude that k = k._ = J

l; = 3 In the case

i'. , i s f i xed , the s o l u t i o n of the s e t of 'jquutionc; c jn s ido red i s unique.

P u t t i n g kj=ll we obta in eqs . ( 3 5 a , b , c ) . Cor.so'ju^t.tly, the corresponding

A- inequa l i ty i s ex t rema] . The A can be found fr-jm eqs . (21) and ( 3 5 ) , butB

the genera l express ion i s not very i l l u m i n a t i n g .

2) Af ter having d i scussed t he se genera l types of extremal i n e q u a l i t i e s we

now turn t o a s p e c i a l c a se , v i z . | A | =C; let. A = { 1 , 2 , 3 , ^ , 5 ) 6 } . In t h i s

c a s e , i t i s convenient t o choose for j>'(A) the s e t of a i l even subse t s

B of A such t h a t | B | <_2. Using t h e n o t a t i o n k. . = k , . . , we can w r i t e the

eqs . (28) as

k. . >_ 0 for a l l p a i r s i , j such t h a t 1 <_i< j<^6

k . . + k , „ + k > krf for a l l permutat ions ijk£mn of 123^56i j k£ mn — 0 , ., * . , , . ,

s u c h t h a t i < j , k < i t , m < n ; i < k < m .

(39a)

(39b)

(39c)

In this case, there is exactly one extremal A-inequality (up to a

permutation of the vertices) of each one of the three general types derived

under i(a), (b), (c). Searching semi-systematically we have, in addition,

found three special extremal inequalities (again counting cases which

differ only by a permutation of the vertices as a single case). The way in

which they have been arrived at suggests that there are no more extremal

inequalities in this case. We hope to present a more complete analysis in

the future.

The six extremal inequalities are represented in Tables 1a and 1b. Table

1a gives the values of the ko, BeJ"(A), Table 1b those of the X,,Be D

(multiplied by a common factor in order to avoid fractional values). The

first columns label the type of inequality, the last column of Table 1b

gives the number II of distinct extremal inequalities obtained from the one

represented by a permutation of the numbers 1, 2, 3, h, 5, 6.

Table 1a. Values of the k_» B ê / ' ( A } , for extremal A - i n e q u a l i t i e s in the

c a : e A = {1 ,^,l,k,<},é}.

a

b

Q

d

e

f

k0

0

1

o

1

1

1

k 1 2

1

0

Ü

1

1

0

k 1 3

0

0

0

1

0

0

Sk

0

0

0

1

1

1

k 15

0

0

0

1

1

1

k i 6

0

0

0

1

1

1

k 2 3

0

0

1

0

0

0

k 2 .

0

0

1

0

1

1

k 25

0

Q

i

0

1

1

k 26

0

0

1

0

1

1

S>

0

1

1

0

0

1

k35

0

1

1

fl

J

1

k36

0

1

1

0

0

1

\ 5

0

1

1

0

0

0

Sc

0

1

1

0

0

0

Sc

-J

1

1

0

0

0II

Table 1b. Values of the A_, Bfc^'(A), for cxtroraal A-inequalities in then e

case A = {1,2,3)^55,6 } , The last column gives the number }! of distinct

extremal inequalities obtained from the one represented by a permutation of

the numbers 1,2,3,^,5,6.

• a

c

d

e

f

I

1

- 1

- 1

0

1

A12

1

7

1

- 1

- 1

- 1

A13

- 1

1

1

- 1

0

- 1

hk

- 1

1

1

- 1

0

1

A15

- 1

1

1

- 1

0

1

A16

- 1

1

1

- 1

0

1

A23

- 1

1

0

1

0

- 1

hk

- 1

1

0

1

0

1

A25

- 1

1

0

1

0

1

A26

- 1

1

0

1

0

1

A3l

1

- 1

0

1

1

1

A35

l

- 1

0

1

1

1

A36

1

- 1

0

1

1

1

1

- 1

0

1

0

- 1

AU6

1

- 1

0

1

0

- 1

A56

1

- 1

0

1

0

- 1

ÏI

'i 5

15

6

6

60

60

The extremal A-inequality labeled a is, of course, one of the type

derived by Sherman, the one labeled c is a special case of Newman's

inequalities, referred to in section )*; the other ones are new, as far as

we know.

Remark: One can prove that the inequalities (c) and (I7d) are the only

Newman inequalities which are extremal A-inequalities.

References.

1) A.J. Goldman and A.W. Tucker, in: Linear Inequalities and Related

Systems, H.W. Kuhn and A.W. Turker eds. (Princeton University Press,

Princeton,1956) p. 19 ff.

2) S. Sherman, Comm.Math.Phys. I M 1 9 6 9 ) 1.

3) J. Ginibre, Phys.Rev.Lett. 23. ( I969) 828.

I4) C M . Newman, Z. Wahrscheinlichkeitstheorie verw. Gebiete .33 (1975) 75.

5) R.B. Griffiths, in: Phase Transitions and Critical Phenomena, Vol. 1,

C. Domb and M.S. Green eds. (Academic Press, London, 1972) p. 72 ff.

6) J.L. Lebowitz, Comm.Math.Phys. J35 ( 1971*) 87.

7) C.A.W. C" tteur and P.W. Kasteleyn, Physica 68_ ( 1973) 1*91.

70

IV. CORRELATION-FUNCTION IDENTITIES FOR GENERAL ISING MODELS

Abstract.

For Ising models with general interactions correlation-function

identities of the form 1-r.X^ ( o_ > < a_c. > = 0 are studied. It is shown that

each identity of this form is equivalent to a set of special identities, in

which the coefficients A take only the values - 1 , 0, 1. A necessary and

sufficient condition for the validity of such a special identity is derived.

71

1. Introduction.

1) 2)In two previous papers, to be denoted hy I and II , a class cf

correlation-function identities for Ising models with pair ii.teractions was

studied. These identities have the form

ÏBCA

(1)

where X , for all B, is independent of the interaction parameters of theD

system, and they hold for all values of these parameters. In II a necessary

and sufficient condition was derived under which the identity (1) holds. It

turns out that this condition consists in a simple set of linear equations inthe ^„'s, with coefficients 0 and 1.a

In this paper we shall show that if, for a given Ising model- with

general n-spin interactions, one has an identity of the above-mentioned

form, the left-hand side of eq. (1) can be written as a linear combination

of special functions, each of which is zero for all values of the

interaction parameters. These functions have the same form as the left-hand

side of eq. (1), but the coefficients A_ take only the values -1, 0, 1. Vie

then proceed to give a necessary and sufficient condition under which these

special functions satisfy the identity (1).

As an application, it will be shown that ii. the case of Ising models

with only pair interactions this condition is equivalent to that of II,

Theorem 1 (applied to the special functions mentioned). We also study in

some detail the equation < ax0Y ' ~ ( ax ' ( aY * = °" Finally> it will be

shown that identities of the special type mentioned are readily constructed

for arbitrary Ising models.

2. Definitions and notation.

Let V be a finite set of elements called vertices and -''(V) the family of

subsets of V. Elements of f(V) will be denoted by A, B, C,..., and subsets

of 'J (V) by a, ß, y except a specially chosen subset (and some derived

sets), called the bond set, which will be denoted by A . We give ^(V)

the structure of a group by defining the product AB of two elements A and B

of ? (V) as the symmetric difference ( A \ B ) U ( B \ A ) . The unit element is the

empty set and every element is of order two.

72

v) we define

V = U A , P = IT A ,a a „_ (2)

•„•here V 0 = P 0 = further g(a) denotes the smallest subgroup of j (V)

having a as a subset.

A set a C.'(v) will be called a cycle if P = 0, and cycle-free if

there is no ß C u , gjiø, such that Va = 0. Observe that by these

definitions the set 0 is both a cycle and cycle-free, whereas the set {0} is

a cycle and not cycle-free.

If A e (V) and 6 cy'(ï), a set a C g will be called a factorization of A

with respect to B if o is cycle-free and P = A; the elements of a will be

called B-factors of A. A„ denotes the set of all factorizations of A with

resoect to B.and B. the set of all ß-factors of A. Obviously, A is notA ß

empty if and only if Aeg(fi).

For a given finite set V and a given set 2'1?{V) we define an Ising

model on -Ò as a pair W>, K) where K is a mapping from -Ä into the complex

numbers. Let J be the set of all mappings o:V -+ {1,-1}. The Hamiltonian

of ( -A.K) is defined by

K ( 0 ) = - I KX (3)

where a is defined byX

aX = " avvex

Jò is called the bond set of ( Ji,K) and K the interaction parameter

associated with the set X. The normalized and unnormalized correlation

functions (o.),A and < a. > ,i „ , respectively, are, for a given set

A 6 <?(V), given by

-H ,, „(a)

as y (5:

where Z , , the canonical par t i t ion function, is defined by Z ^ = (1) „ „.

We have taken kT = 1.

If for a given interaction function K on Ji, K„ = 0 for a given X e J 3 ,

and J' =^i\{X} and K' is the restriction of K to Ji', then

Henceforth, we shall omit reference to K. As in II we shall work

exclusively with unnormalized correlation functions, the translation to

identities for normalized correlation functions being trivial.

3. General formalism.

We first prove a simple lemma about (c. hi •

Lemma 1 • If V is a finite set, £c>?(V) and A £ ^ ( V ) , then (°A)ja, = 0 for

every Ising model on A if and only if A

Proof. Since for any

Vx

where cv = cosh K and s = sinh K , we can write

(7)

Ißc

ƒJi }x x'eß

sx' (VPJØ • (8)

From eq. (8) one easily derives that (a.)s = 0 for every Ising model on

if and only if (o«Cp )w = 0 for all BC-9, i.e., because of the relation

(9)

if and only if A ^ P for all ßC^, which proves the lemma.

Let us now, for a given Ising model (i/3,K), consider the function

(10)BEg(a) D

where Aej>(v), JCJ>(Ï) , and where, for all B, Xg is an arbitrary complex

number not depending on the interaction parameters of the system. By Lemma 1

we may, without loss of generality, assume aCg(-ô) and

We shall study the following identity

= O for all Ising models on v/3. (11)

For this purpose we first apply to the right-hand side of eq. ( 10) a

Fourier transform with respect to the group g(a). Since g(a) is abelian and

of order two, its characters satisfy x(B)6{1,-1} for all Beg(a). We can

construct the characters, labelled by the elements of g(o), in the

following way. Let a' tie an arbitrary but fixed maximal cycle-free subset of

a (i.e. a minimal generating set of g(a) ) and Ceg(a). Further, let y C o '

"be the unique factorization of C with respect to a'. We define for 3 e a'

1 if B£a'\y

and for B Sg(a)\a'

C K B C(13 )

where $ is the unique factorization of B with respect to a'. It is easy to

see that

Xc(B)xc(B') = XC(BB') (lit)

and hence that \ is a character of g(a). Observe that the labelling of the

characters depends on the choice of a'. The orthogonality relation for the

characters reads

1I XJB)X„,(B) =

|g(a)| BSg(o)

Finally we have X-g(C) = XC(B) •

If we define

and

I Xc(B)(oBEg(a) C E

Xp(B)A.B

" Beg(a)

we can write Aatô|A) as follows:

A°(A|A) = I CA^U|A) .|g(a)| CSg(a)

We now have the following theorem.

(15)

(16)

(17)

(18)

75

/

Theorem 1. Let V be a finite set, J4c,?(v), A£g(J) and aCg(^). If

A (jft | A) =0 for every Ising model on i , then there exists a yCg(a) such

t.-iat A^(A|A)=O for every Ising model on Jb for all CEy, and *c=0 for all

CEg(a)\y . Furthermore, 0 £ Y -

Proof. Since Aa(./>|A)=0 for all K, we have

— — Aa(vÄ[A) = 0 for all K and all xej,3K

from which it can easily be seen that

B£g(a) B B X '• B A X J

Repeating this argument, we find that eq. (19) holds for all X E g M ) , and in

particular for all X£g(a).

Multiplying eq. (19) by xc(X), with Ceg(a), using eq. ( 1U) and summing

over all X£g(a) we find

I. . I. . Xn(Bx)V(B)VaRYlA(amraA).A = ° • (20)

Using the fact that g(a) is closed under multiplication we find from eq. (20)

that

C C= 0 for all C€g(a). (21)

The first part of the theorem follows directly from eq. (21).

From eq. (lit) it follows that xß(0)=1 for all BGg(a), and that hence

Now suppose 0 Ey, i.e. A"(Ji|A)=O for all K. Let ß be a factorization of A

with respect to A . It follows that for X e 8

(a a ) (o a ) + (a )Jo a a )Beg(a)

_ 2 n(a ,{X}) ^> (a ) (o o o ) - 0BSg(aU{X}) B v ? l B A X J 5

' / ne re , for 6 C J 5 > n ( a , 6 ) = \S\ - r(aUS) + r ( a ) , wi th r ( a ) = | a ' | for any

maximal c y c l e - f r e e s e t a'Ca (which i s e a s i l y seen t o be independent of

(22)

the choice of a') The factor £ '» arises from the fact t.iat if

XSg(a), we have to take into account a factor 2. From eq. (??) it follows,

that if we repeat the argument for all X£ S we find

B€g(aUB)a a a_B A F BSg(aUß)

This would, for any real interaction function, imply that (ag)i=0 f o r

B 6 g ( a U g ) , which is impossible by Lemma 1, since a,6CJ}. Hence the

second pajife of the theorem follows. £

From Theorem 1 and eq. (18) it follows that if Aa(yi|A) is zero for all

Ising models on ,#, it is a linear combination of functions l\rU}\h) which

are themselves zero for all Ising models en -A.

• Now every A (*#[A) with C^0 will consist of terms with xr.(B) = 1 and terms

with xr(B)=-1> B,C6g(a). If we introduce, for a given C £ g ( a ) , the set

g(a,C) = {B€ g(a) | xn(B) = 1} , which is easily seen to be a group, and if

D is any element of g(a) with

A"W|A) =

we can write A"(V'Ï|A) as

Beg(a,C)

which is a special case of the following function

i / j I A TI ^Vfc. \ f ,* \ f ^ / \ f

Y Beg(y) i-

where y is any subset of .?(V) and A and D are arbitrary subsets of V. This

kind of functions has already been studied in connection with correlation-

function inequalities. Sherman ^' and Ginibre ' have shown that for any

ferromagnetic Ising model on Ji>

(23)

A (A|A,D) for all and A,D 6

We now proceed to give a necessary and sufficient condition under which

A (J1|A,D)=O for all Ising models on >;j .

Theorem 2.

Let V be a finite set, A , Ï C / ( V ) , and A.DE^V). Then A («£|A,D)=O for

all Ising models on A if and only if

De g(aU Y) for all , (25)

77

In words, condition (c5) expresses that every factorization of A with

respect to ÁÍ) contains a subset which can be supplemented by a subset of Y to

yield a factorization of D with respect to/iUy .

This theorem is a consequence of two lemmas which we shall prove first.

Observe that if Aíglj-O), A, is empty and the identity holds for any D.

Lemma 2. Let V be a finite set, Jh , yCp(V) and A, D S ^ ( V ) . Then

A W | A , D ) = 0 for all Ising models on A if and only if A (Ø|AP ,,D) = 0

for all aC A and all a' C a .

Proof. Using eq. (7) and the fact that c = 1+sfi, we find by a straight-A A

forward calculation that for any xe^}

' |A,D).n(Y,(X})

|AX,D)

(26)

where A' = ?>\{X] and n(y,{X}) is defined as in the proof of Theorem 1. If

we apply eq. (7) to all XsJb and iterate eq. (26) we obtain

A>i|A,D) =aO'3 a'ca X£a X'£et\a'

If A u (0|AP , ,D) = O for all aC-ò and all a'Ca it follows that A W | A , D ) = O

for all Ising models on £> . Conversely, suppose A W | A , D ) = 0 for all Ising

models on A . Vie select an aßj> and consider an interaction function K such

that K = 0 if Xe^\a, Kx^° if X

£ a . From eq. (27) it follows that in that

case

A W|A,D) = o = I n c s n" a'Ca XGa' X'ea\a'

2s ). (28)

If we divide out a factor n CVSV '

w e obtain a polynomial in the variables

t^ = cx s , X6a. This polynomial is (by continuity) zero for all values of

the variables K ., X€a, even if K^^c,^^ = 0. We select an a'Ca and consider

the restriction K' of K to a. We now take K"=0 if Xea' and K ^ K ^ O if

Xea\a' and obtain a polynomial in tanh K_ of which the coefficient of the

term of highest degree is A (0|AP ,,D). Since this polynomial is zero for

all values of KQ, it follows that A Uc((0lAP

T>D) = 0. Since a and a' were

arbitrary, the lemma follows. I

T«

Lemma 3- A. ( 0 | A , D ) = 0 for a given set YCJ?(V) and A,De^(V) if and only if

A=0 implies

Proof. Using eq. (9) we find

i 'Vi'»,\(»|A,B> •

The lemma directly follows from eq. (29).

(29)

Proof of Theorem 2. From Lemmas 2 and 3 it follows that A (-"|A,D) = 0 for

given A,Dei"(V), Y> ^

relation AP ,=0 implies D6g(yUa) where ot'CaCgC^). From this it follows that

and al l Ising models on /\ if and only if the

'CaCgC^). From this it follows

if and only if for a l l aClsuch thatA (VT.|A,D) = 0 for all Ising models on y

AGg(a), D is an element of g(aUy). The theorem follows from the last

statement. •

Theorem 2 can be used ir. several ways. First, it states that for fixed Ji,

Y and A there is an identity of the form A (;3|A,D) = 0 for every set D in

the intersection (to be denoted by g.(y)) of all sets g(aUY) with aeAjj . It

should, however, be observed that not all these identities are distinct, and

that some of them are of the

then it is easy to see that

that some of them are of the form 0=0. Suppose D,D'6g.(Y) and DD'Sg(YU{A}),

A ^ ( A | A , D ) = A (^|A,D'); in this case D and D'

will be called equivalent. Furthermore, if D=0, terms of opposite sign in

A (-A|A,D) cancel pairwise so that the djntity holds trivially.

Alternatively, it follows from Theorem 2 that for fixed ^3, A and D there

is an identity of the form A (y}|A,D) = 0 for every set YC'"'(V) having a non-

empty intersection with all sets {BEJ"(V) |B=DX for some X6g(a)}. Obviously,

if A (~*|A,D) = 0, then A ,(V3|A,D) = 0 for every Y'^Y-

h. Applications.

1) As a first application we consider the case g(y) = (0}> i.e. we

study the equation

[30)

79

The equation has been studied by SetS for some special cases of X and Y in

ferromagnetic systems Setô found, that for a ferromagnetic system with

pair interactions and in a sufficiently general magnetic field

( cyj,, ) = ( o > < 0 o > and < 0^,0,0^ ) = < <i^-, > < i^v^ > if anr3 only if

spin 1 "decouples" from spin 2 in the former case and spins 1 and 2

"decouple" fron spins 3 and it in the latter case. We shall nov generalize

these results to Ising models with arbitrary (not necessarily ferromagnetic)

n-spin interactions. If X or Y=0 eq. (30) is trivially true; furthermore,

if XY=0 and X^ø, Theorem 2 tells us that e-j. (30) can not be valid for all

Ising models on A. So we assume X, Y, XY ^ 0 and furthermore X, Yeg(.^). We

have the following lemmas.

Lemma k. Let V be a finite set, J9c?(y) and X, Y6g(~i).

statements are equivalent.

(a) For all yCJb, XYeg(y) implies Xe g( Y).

rhe following two

(u) = 0.

Proof. (a)->(b). Suppose ^P\ t 0. Then there are oeX^, 06ïjj with common

-'3- factors , i.e. there are sets ct'Ca, a'^a and g'Cß, ß #ß, such that

P \ , = P ^ ,. Let a' and ß1 be such sets with the property that |a\a'| is

maximal. We have

= Pa,P3, (3D

and hence XYeg(a'Uß'). It follows from (a) that Xeg(a'uß')> i.e. there are

sets ot"cct', ß"cß' such that X = P „P „, and hence Y = P , »pßi\ß'f 1*

follows that

Pa\,'Pa'\a" = P ß " •

Since a is cycle-free, ¥„,é since ß is cycle-free, P ,. „ 4- 0. This

leads to a contradiction because |a'\a"| has been assumed to be maximal.

Therefore, BvnB = 0.

A I

("b)-v(a). Take any cycle-free Y such that XY = P and let ßSY^; then

Since there is a y'CyXß such that P . = P , and

hence X = "P, . ,. ,. Since (Y\B)\Y'CY, this completes the proof of the

lemma. •

The following lemma tells us something about the structure of A,.

Lerna 5. Let V be a finite set, X£?(V), :#0,and .Ae.?(v). Then every

"££(&..) belongs to a factorization of X with respect to *3 U{Z},i .e.,there isA A

a Y C A , such that ZP =X and there is no y'Cy such that Z=? ,.

Proof. Let 3Cu!>,,such that Z=PfJ. It follows that (v3v)7 = (v-i,) z ° M,) y 5s 0,

and hence ,by Lemma '4, applied to •$,, instead of A, th°re is a Y C A . such

that ZX6g(y) and Zíg(Y),from which the lemma follows. |

Leirma 6. Let V be a f in i te set , X.YeJ'tV) and AC J>( V) ;then »*3,O V = 0 if1 A I

and only if g(^ x )ngC/\,) = {0! •

Proof. If = {0} , then trivially x n ^ = 0. Suppose now

Jiy^Aj, = 0 but gWx)ng(.ftY) jí {0}, i.e., there are sets j C ^ y , 3 C ^ ,

a,S # 0, such that 0 4 P = P geg(^ Y). From Lemma 5 it then follows that

there is a Y CÆ- such that PgP = X,and there is no y'Cy such that Pg = P , ;

hence ®nj^x * ^' wilicl1 l e a d s "t° a contradiction and ends the proof of

the lemma, f

Lemma 7- Let" V be a finite set, X.YeJ'iV) and JJCJ!(V). If J3..'1-3y = 0 and

A' =A\(,3XU/>Y), then g W ' l n g t ^ U ^ ) = {0}.

Proof. Suppose g(£' )ng(^x U-^) i {0}, i.e.,there is a [ZEgW ), ctC3.(,

gcA,, a,ß # 0, such that Z = P Pn. [t follows that ZP. = P Sg(A,). ByX 0[ p 0 Ct A

Lemma 5, there is then a Y C/J>r such that ZP P = X, and there is no Y'CY

such that ZPg = P ,. Since Z?g(j\() Ug(jJï) it follows that ßn,ä ^ 0,

which leads to a contradiction. ^

From Theorem 1 and Lemmas h and 6 we immediately obtain the following

theorem.

Theorem 3- Let V be a finite set, JiC?(v) and X.YSgtø), X,Y jt 0. Then

ojKia a ) e v ery Ising model on JJ if and only if

= {Øl-

From Theorem 3 together with Lemma 7 it follows that if

81

\

o a ) = (ov),(a„)o the bond set is the union of three pairwise

disjoint subsets , -8y and 3V = A\ (JX.U- such that gUx)ngliJy) = {0}

and g U ' ^ g t L U ^ ) = {0}. It follows that one can extend -ft with allA J.

elements from gC^,.), g(3v) and g(J>') to a new bond set 5V' with the propertyA 1

that (a A], „[a a ).„ = (a ) „(a ),„.

2) In our second application we consider Ising models with pair inter-

actions only, i.e., with 2c ? (v) : = [S£?{v) I |B| = 2}. This case was treated

in I and II. We shall show that Theorem 1 of II, restricted to functions

of the type (21t), is equivalent to Theorem 2 of the present paper,

restricted to •AC? (v); by Lemma 1 and Theorem 1, the corresponding

identities are the only ones that need to be considered.

With each Ising model with ÍC? p(V) we can associate a graph G = (V,-/3)

with vertex set V and edge set J>. Let ACV, |A| even. G defines a set of

partitions of A, denoted by II (G), in the following way. If IT is a

partition of A, i.e. a family of pairwise disjoint non-empty subsets of A

whose union is A, then n6Il.(G) if there is a set goisuch that two elements

are in the same block of i if they are in the same connected component of

the spanning subgraph Go = (V,B) of G. Conversely, every spanning subgraphP

Gg of G defines a partition of A, which will be denoted by ir(A,Gg).

The following lemma was used in the proof of Lemma 3 of II:

Lemma 8- Let A£-?(V) and gC/j' (V). If n(A,G.) is even, there is a cycle-free aCg such that A=P .

a

Proof. Take aCg such that IT(A,G ) is even and |a| minimal. Then a is cycle-

free and each B£a separate, two odd subsets of A in G . It follows that each

vertex in A (in V\A) is incident with an odd (even) number of elements of a,

i.e., A=Pa. H

Let us define, for any set BEJ >(V) and any partition i of a set X£/(V),

'1 if the number of elements of B in every block of TT is even,

0 otherwise.

Furthermore, we define n (A,G)A

following proposition.

(G)|n (A) = it. We now have the

82

Proposition. Let i c y ^ v ) , yC?(v), A £ g W ) ,

statements are equivalent:

(a) D£g(aUy) for all aGAjj ;

The following two

(b) / J n ^ = 0 for all nenv(A,G) . (33)

Proof. (a)->(b). Let iten,,(A,G) and1 Vsuch that Tt(V,G.)=n. Since n(A,G.)

is even there is, by Lemma 8, a cycle-free aC3 such that A=P . Hence,

aeA,, and hence, by (a), Deg(aUy), i.e., there are sets a'Ca, Y ' C Y suchA,eachthat D=Pa,P ,. Let IT '=TT (V,G ) ; then n^.ÍDP ,) = nit,(P((1) = 1 because

component of G contains an even number of vertices of P , . Since it' is a

refinement of IT, we also have ,i (DP ,)=1, and iiencp (BDP ,) = ^

every B£g(y). Since P ,6g(Y)> we conclude that

for

I n (BD) = I n (BDPvi:Beg(Y) ïï — '-1 *

I(which proves (b).

(b) -»(a). Let a£AA and it=ii(V,G ); n6H,.(A,G) because A=P . Since n (0)=1i«Æ a V a IT

we have £ B g , )rl1,(

B) — 1' a n d hence, by ( b ) , £ B e , .n^fBD) ^ . 1 . So there

is a Beg(Y) such that n (BD) = 1, i.e., BD is partitioned by it into even

subsets. In other words, it(BD, G ) is even. By Lemma 8, BD=P , for some

a'Ca, and hence D=P ,Beg(aUy). I

It is easixy seen that the set of equations (33) is equivalent to any

set obtained from (33) by replacing V by X, where V UAUDCXCV. This

suffices to show that the proposition establishes the relation between

Theorem 2 and Theorem 1 of II referred to above.

A convenient way to find, for a given <AC-?(v) and AjDe-^V), a family of

sets Y such that A (A|A,D) = 0 for all Ising model on 3 is the following.

Consider the set n (A,G), ordered by refinement. Let n*"(A,G) be the set

of minimal elements of this ordered set, aCA •, and it a refinement of TI(A,G )</} a

such that it£II„(A,G). Since every two points v ,ir which are in the same

block of it are connected in G by a chain, {v ,v„}eg(oi). Using this fact it

is easy to see that g(it)cg(a) and that, hence, if ycJ> then g(nUy) c g(aOy) •

It follows that

g(nUy) C n g(aUy) ,7ieHm(A,G)

so that if Y is such that DGt)>, then A (2>\A,D)=0 for all Ising models on >3.

83

Since i|i can be found relatively simply ("because it does not depend on

•-he detailed structure of the graph) examples of identities of the form

A (j)\n,D) = 0 can easily be constructed from the condition DE'JJ. For example,

y,,v ,v-} a boundary sequence of Glet G be a planar graph and {v ,vo,v ,y,

(for the definition of a boundary sequence, see I). Let A =

and D = {v ,v }. The set cf minimal partitions consists

12 | 3 | L>6), n;) = (:3| V | l O ( ^ \ " \ % )

iv.,vo,v ,Vj ,v ,v^}

in fnii; case of = " 3 =

T^ = (25|3l)|i6) and TT = ( 12 | U 5 | 30). If we nov; construct the sets

$. : ={BCA|B = DX for some X6g(ïï. )} (each of which consists of eight

elements), then the sets y such that Yn<f>' ^ ø f ° r 3-11 i are readily

constructed.

3) As a last example we consider a set of special identities which can

readily be constructed for Ising models with n-spin interactions.

Let V be a finite set, .'!>C J\v). Suppose that V is the union of three

pairwise disjoint sets V , V and V and that ,/> - jiuá <j2>, with

•3c?(vuV )\?(V ), BcJ(VUV )\-?(V ) and u}Ci(V ), i.e.,there is no Be AI I S 3 ^ i ^ S 5 5 3

containing both a vertex of V and a vertex of V_. We further assume that

Consider now non-empty s e t s

X Xeg(-A). Take aCji such t h a t X

such t h a t X-lnV2 = 0 = X/1'/-, and

and l e t a. = a ^ . ( i = 1 , 2 , s ) . Then

we have X,P1 t

= X„P P2 ap

Since X,P '">v„ = 0 and X„P P n v . = 0 we have

P CV ,or X^giaUi/XV ) ) . By Theorem 2,we t h e r e f o r e have the i d e n t i t y

sei(vs)= 0 (3k)

That separating sets are readily constructed can be seen as follows.

Consider any set A CV. Let A be the set of all vertices u, u$A , such

that there is a set BEjb containing u and a verte:, v in A.. Let

A 3 = V\(A.UA ). Then any V DA such that A \V and A \ V are not empty is at- I s S S I S C- S

separating set.

As an illustration, consider the following example. Let v ,v£V and

define -Ä = ?(V) \{v1 ,v„}. The set V = V\{v ,v„} is a separating set and

we have the identity

8I4

where j' (V ) and 2 (V ) are the families of even and odd subsets of V ,e s o s s

respectively.

It can easily be seen that eq. (35) can be written as

I (-D-=J (v)

,vo |(35')

a relation which was already derived in II. It is clear that eq. (35') re-

mains valid, if we extend the bond set with those 3e.?(V) which do not

contain both v and vo.

Concluding remarks.

1) At first sight, an obvious generalization of the identity A (/5|A) = 0

would be an identity of the form

(36)

with A and A' arbitrary subsets of V.

If A,A'eg(j"J), it is clear from Lemma 1 that we can restrict ourselves

to the case g(a)cg(.3). In that case, however, one finds by the use of

multiple differentiation (see the proof of Theorem 1) that eq. (36) is

equivalent to the identity A (J1|AA') = 0 . If A and A' are not both elements

of g(jj), there can only be non-vanishing terms (°B°a i/3^aBaA'^ """*" AA^ßto);

furthermore, a term is nonzero only if B is of the form B=AC with Ceg(^),

and it is easy to see that eq. (36) again reduces to an identity of the

form Aatø|AA') = 0.

2) Lemma 1 tnd Theorem 1 can be combined to the statement that every

function of the form

r , / \ / \ i -}~, \

with A, yC?(V) and AGg(yJ), which is zero for every Ising model on 4, can

be written as

B€Y\g(3)

|A) (38)

85

where and ^T6U>|A) are defined by eqs. (16) and (1?) with A =0 for

Beg(yUg(jl) )\y> and where each individual term is zero. This shows that

Theorem 2 together with Lemma 1 covers the most general case of an identity

of the form (11).

References.

1) J. Groeneveld, R.J. Boel and P.W. Kasteleyn,to be published in Physica.

2) R.J. Boel and P.W. Kasteleyn, to be published in Comm.Hath.Phys.

3) S. Sherman, Comm.Math.Phys. J_ii ( 1969) 1.

It) J. Ginibre, Phys.Rev.Lett. £3 ( 1969) 328.

5) K. Setô, Prog.Theor.Phys. £§. (1976) 683.

SAMENVATTING.

In dit proefschrift worden bepaalde relaties tussen spincorrelatie-

functies in Ising modellen onderzocht. Deze relaties worden bepaald door

een klasse van functies (gemakshalve A-functies genoemd) die gedefinieerd

zijn als lineaire combinaties van Produkten van twee spincorrelatiefuncties.

De coëfficiënten (aangegeven met A's) van deze produkten zijn willekeurige

complexe getallen die niet van de koppelingsparameters van het systeem af-

hangen.

Onderzocht wordt wanneer voor een gegeven Ising model (gespecificeerd

door een verzameling punten die de spins voorstellen en een verzameling

lijnen tussen punten, die aangeven tussen welke spins er wisselwerking is)

een gegeven A-functie

(a) nul is voor alle waarden van de koppelingsparameters (de relatie die

hierdoor bepaald wordt noemen wij een A-identiteit),

of

(b) niet-negatief is voor alle niet-negatieve waarden van de koppelings-

parameters (de relatie die hierdoor bepaald wordt noemen wij een

A-ongelijkheid).

Het blijkt dat een nodige en voldoende voorwaarde waaronder (a) geldt, be-

staat in een stel ho-nogene lineaire vergelijkingen in de bovengenoemde

coëfficiënten X, met coëfficiënten 0 en 1. Een nodige en voldoende voor-

waarde waaronder (b) geldt wordt gegeven door het corresponderende stel

homogene lineaire ongelijkheden. De coëfficiënten 0 en 1 hangen nauw samen

met bepaalde connectiviteitseigenschappen van een groep punten in de graaf

die men op een natuurlijke manier aan ieder Ising model met paarwisselwer-

kingen kan toevoegen, en zijn eenvoudig te bepalen.

In hoofdstuk I wordt een voorbeeld gegeven van een klasse van A-identi-

teiten, die gelden voor Ising modellen die gedefinieerd zijn op planaire

grafen.

In hoofdstuk II worden bovengenoemde nodige en voldoende voorwaarden

afgeleid en worden enkele voorbeelden van A-identiteiten en A-ongelijkheden

gegeven.

Hoofdstuk III bevat een meer gedetailleerde studie van de A-ongelijkhe-

den. Een van de resultaten uit dit hoofdstuk is dat iedere A-ongelijkheid

(uit een bepaalde klasse) te schrijven is als een positief lineaire combi-

natie van een vast eindig stel "extremale" A-ongelijkheden (uit dezelfde

87

klasse). Deze extreraale A-ongelijkheden geven, binnen de beschouwde klasse,

de scherpste boven- en ondergrenzen van spincorrelatiefuncties in termen van

correlatiefuncties van lagere orde. Voorbeelden van extremale A-ongelijk-

heden worden afgeleid.

In hoofdstuk IV worden A-identiteiten onderzocht die gelden wanneer er

n-spin interacties (n >2) in het Ising model aanwezig zijn. Het blijkt dat

iedere A-functie, die een A-identiteit bepaalt, geschreven kan worden in

termen van eenvoudiger A-functies (waarin de bovengenoemde coëfficiënten

, -1, 1 of 0 zijn), die ieder voor zich een A-identiteit bepalen. Een

nodige en voldoende voorwaarde wordt afgeleid waaronder voor deze eenvou-

diger functies (a) geldt. Enkele toepassingen worden daarna behandeld.

37UDIE0V1.RZICHT.

-7 mei Iyo8:

september 1968:

2~J januari 1971:

begin 1973:

1 januari 197't:

7 mei 197)4:

eindexamen HBS-B; Het Goese Lyceum, Goes,

aanvang natuurkundestudie, Rijksuniversiteit Leiden,

kandidaatsexamen natuurkunde en wiskunde met scheikunde,

experimentele stage in de werkgroep van Prof.dr.ir. N.J.

Poulis.

indiensttreding bij de Rijksuniversiteit Leiden,

doctoraalexamen natuurkunde met bijvak wiskunde en de aan-

vang van het promotieonderzoek onder leiding van

Prof.dr. P.W. Kasteleyn.

In 1 nyI4 nam ik deel aan de iluffic zomerschool te Wageningen, in 1975 aan de

IUPAP conferentie te Boedapest en in I976 aan de zomerschool over kritische

verschijnselen en faseovergangen te Banff (Canada).

Het proefschrift is getypt door mevrouw S. Helant Muller-Soegies (waarvoor

mijn onuitputtelijke dank). Het eerste hoofdstuk van dit proefschrift kwam

tot stand na intensieve en langdurige samenwerking met Dr. J. Groeneveld

(Rijksuniversiteit Utrecht).

Once, when T.A. Edison showed one of his new inventions, somebody said

to him: "It's marvellous, Mr. Edison, but what is the use of it?"

Edison answered: "And what is the use of the newborn child?"

90