Limit theorems for sums of dependent random variables occurring in

Z. Wahrscheinlichkeitstheorie verw. Gebiete 44, 117-139 (1978)

Zeitschrift ft~r

Wahrschein l ichkei t s theor ie und verwandte Gebiete

�9 by Springer-Verlag 1978

Limit Theorems for Sums of Dependent Random Variables Occurring in Statistical Mechanics

Richard S. Ellis 1 . and Charles M. Newman 2 **

1 Department of Mathematics and Statistics, University of Massachusetts Amherst, Massachusetts 01003, USA 2 Department of Mathematics, Indiana University, Bloomington, Indiana 47401, USA

Summary. We study the asymptotic behavior of partial sums S, for certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number 2, and a positive integer k so that (Sn-nm)/n ~-1/2k converges weakly to a random variable with density proportional to exp(-2[s]Zk/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.

I. Introduction

Suppose {Xj:j = 1, 2 . . . . } is a sequence of identically distributed random variables

with E(IX 1[) < oe and E(X 0 = 0. We define S, = ~ Xj. If the XSs are independent j = l

and E {X~} < o% then the central limit theorem states that S , / I ~ S as n--, oe (in the sense of weak convergence of distributions), where S is normally distributed with mean zero and variance E {X~}. There are two distinct ways in which the assumptions on the XSs can be weakened to render the conclusions of the central limit theorem invalid. In both cases one finds that under suitable assumptions there exists a number re ( l ,2 ) so that as n ~ 0%

n • / - - + S (in distribution); (1.1)

S is a nontrivial random variable with distribution depending only on v and one

* Alfred P. Sloan Research Fellow. Research supported in part by a Broadened Faculty Research Grant at the University of Massachusetts and by National Science Foundation Grant MPS 76-06644 *~ Research supported in part by National Science Foundation Grants MPS 74-04870 A01 and MCS 77-20683

0044-3719/78/0044/0117/$04.60

118 R.S. Ellis and C.M. Newman

other scaling parameter, but otherwise the distribution of S is independent of the particular nature of the Xj's.

In the first modification away from the central limit theorem, the Xj's are kept independent but have infinite variance. Then one must choose v < 2 because of the high probability of large summands in S,. This situation is completely understood [9]. Given that appropriate assumptions on the tails of the distribution of X 1 are satisfied, then the limiting S exists and has a (symmetric) stable distribution of exponent v. The infinite variance of X1 is mirrored in the infinite variance of S.

The second situation in which non-Gaussian limits can arise occurs when X 1 and S have finite variance but the Xj's are strongly positively correlated. Here one must choose v <2 because of the high probability for reinforcement among the summands in S,.

It is believed that the second phenomenon is typical of various models which are studied in statistical mechanics; in particular, of(ferromagnetic) Ising spin systems. These systems are aggregates of stationary, positively correlated random variables {X}n): j = 1,..., n} (for n = 1, 2,...), which represent the individual spins in a lattice model of a magnet. The triangular array structure, which is natural in these systems, can often be eliminated by first taking the limit n ~ oo before dealing with the partial sums; this elimination is not possible in the particular systems treated in this paper. The degree of positive correlation among these random variables is controlled by a positive parameter/~, proportional to inverse temperature. An important feature of certain Ising systems is the existence of a critical value/?c of/?. For/~(0,/~c), the individual spins are weakly positively correlated, and the limit laws of classical probability theory are valid. For/~(flc, oo), the correlation among the spins is strongly positive in a fundamental sense. This change from weak to strong correlations as/3 increases through/?c is an aspect of a phase transition of the Ising system at/~--/?c.

The phase transition is mirrored probabilistically as follows. In the region/~ </~ of weak dependence among the X}")'s, a standard central limit theorem for S n is valid [12]. However, for/~=/~c, physical arguments have been advanced which suggest that a limit as in (1.1) is valid for some w(1, 2). In statistical mechanics v is known as a critical exponent. (For/~ >/3~, the situation is complicated by the fact that due to the strong dependence in this region, the random variables tend to cluster in several ergodic components. However, by the use of conditioning one may focus upon a single one of these components, and a central limit theorem is again valid [10].)

The existence and nature of the limiting S in (1.1) are open problems in almost all non-trivial Ising systems. We consequently study these limits for a relatively simple class of systems known in the statistical mechanics literature as Curie-Weiss or mean field models [3, 14, 22, 26]. These models have been important physically because their critical phenomena reproduce qualitatively certain features of critical phenomena in more complicated systems ([22, Ch. 6]). However, except for the background discussion of [21], our probabilistic approach to Curie- Weiss models is new. This work, consequently, clarifies the nature of conjectur- ed limit theorems like (1.1) in more complicated systems and also yields a number of results of independent probabilistie interest. For example, because of the relatively simple way in which the mutual dependence of the spin random variables in Curie-

Limit Theorems for Sums of Dependent Random Variables in Statistical Mechanics 119

Weiss models is expressed (see (2.2) and (2.3)), results like Theorem 2.1 can be considered as results concerning large deviations for sums of independent identically distributed random variables.

In Section II we state our main results under special hypotheses, saving for Section III their more general statements. In Section IV we present certain topics in the theory of Gaussian quadrature and the theory of moment spaces which are used together with the work of Section III to prove the results of Section II. In Section V we go into more detail concerning the physics of Curie-Weiss models in order to discuss the statistical mechanics content of the theorems in the main body of the paper.

Our work extends earlier results on Curie-Weiss models of [21] and [5]. In [7] there is a more complete presentation of the relevant background material than could be presented here. Limit results for other Ising systems appear in [1], [4], and [-18]. For a discussion of the relevance of such limit theorems to the general study of critical phenomena, see [15]. Limit theorems in a non-statistical mechanics setting but with scalings identical to our own are treated by [-17] and [25].

Acknowledgements. We thank D. Hayes, H. McKean, L. Pitt, J. Rosen, J. Sethuraman, and A. Stroud for useful conversations.

II. Main Results

In this section our results are stated under simplifying assumptions, first for probability measures on 1R, then for spherically symmetric probability measures on ]R t, tE{2, 3, ...}.

The measure p on IR will always be a probability measure which satisfies

S exp(x2/2) dp(x) < m. (2.1)

(Here and below, all integrals extend over IR 1 unless otherwise noted.) Given such a p, we consider the triangular array of dependent random variables {X}")(p):j = 1, ..., n} (n = 1, 2,...) with joint distribution

- - exp dp (x~), (2.2) Z. 2n j= 1

where

Z , = S exp [ ! x l + ' ' ' + x n ) 2 ] I~I d p(xi)- an 2n j=l

These formulae define a Curie-Weiss model (the parameter /? discussed in the

Introduction has been absorbed into p - see Section V below). Set Sn(p)= ~ XJ "). j = l

We write __j;g(.") for XJ")(p) and S, for S,(p) when there is no danger of confusion. The distinguishing feature of these random variables is the way in which their

mutual dependence is expressed. Let I11, Y2,... denote a sequence of independent

random variables with common distribution p and set T, = ~, Yj. Then for each j = l


n E { 1, 2,...} and each bounded measurable function f : IR"~ IR we may write

1 \ 2 n ! f ( r l . . . . , r . ) .

This implies that the distribution of Sn is

- - e x p dp*"(x), (2.3) Z,

where dp*"(x) denotes the n-fold convolution of p with itself. We denote by N(/~, a 2) (x) a normal distribution with mean # and variance ~z

and write N(x) for N(0, 1)(x). We define moments ( j=0, 1, ...)

I~j(p) = S xj dp (x), (2.4)

0, for j odd, ~j = ~ x j dN(x) for j = 0, (2.5)

( j - 1) ( j - 3)... 5 .3 .1 , for j even.

Our simplifying assumption in this section is that

~exp(sx)dp(x)<exp(~)=_,exp(sx)dN(x) , for all real s ,O . (2.6)

If both sides of (2.6) are expressed as power series in s for s small, then inequality (2.6) implies that there exist a positive integer k = k(p) and a positive real number 2 = 2(p) such that

//j_/~j(p) = SO, for j=O, 1 , 2 , . . . , 2 k - 1, (2.7)

2>0, for j=2k .

A p satisfying (2.7) is said to be of type k and strength 2. Our first result, Theorem 2.1, states that if p is nondegenerate (i.e., not a point

mass), of type k, and of strength 2, then if k = 1, {X)")(p)} is in the domain of attraction of the normal distribution N(0, 2-1 - 1) (where necessarily 0 < 2 < 1) and if k > 2 it is in the domain of attraction of the measure with density proportional to exp(--~.sZk/(2k) !). Theorem 2.2 refines Theorem 2.1 by allowing p to depend on extra parameters which are themselves scaled; this leads to additional limit laws. Theorem2.3 guarantees for each k~{1,2,...}, the existence of measures which satisfy (2.6) and are of type k.

We denote by 6 ( x - x o), x o siR, a unit point mass concentrated at x o; by ', d/d x; and by {Hdx): i=0, 1, ...}, the Hermite polynomials

Hi(x ) = ( - 1) ~ exp ~ - ] ~xTx ~ exp . (2.8)

We write S~ ~ v or S n ~ f ( s ) to denote that as n ~ oe the distribution of Sn converges weakly to a distribution proportional to dv=_f(s)ds. We usually omit normali-


zation constants. Given Y a random variable with distribution a, we write Y,-~ a. If

{p,: n=1,2, . . .} is a sequence of measures, then we define S,(p,)= L X)")(P,), j = l

where the {)(1(. ")(p.)} are random variables with joint distribution given by (2.2) with p replaced by p,.

Theorem 2.1. Assume that p is nondegenerate, satisfies (2.1) and (2.6), and is of type k and strength 2. Then

S,(p) - - -~ 6 (s), (2 .9 )

n

S,(p) ~'N(0, a2(p)), for k = 1, n 1-U2k ',exp(--2s2k/(2k)!),) for k>2,

(2.10)

where

a2(p)-2-1-1 >0. (2.11)

Theorem 2.2. Suppose that p is nondegenerate, satisfies (2.1) and (2.6), and is of type k > 2 and strength 2. Then given any real numbers 21 .. . . . 22k-1, measures p, can be chosen satisfying (2.1) so that p, ~ p and

S,(p.) ~exp - 2 ~ 2; (2.12) n 1-,2

�9 j = l �9

Theorem 2.3. Given k6{1, 2, ...}, there is a universal upper bound on 2 appearing in (2 .7 ) (and consequently in ( 2 . 1 0 ) ) :

2<k! (2.13)

For each 2E(0, k !), there are uncountabIy many distinct p's satisfying the hypotheses of Theorem 2.1 for that k and 2. However, there exists a unique measure p(k) which satisfies (2.7) with 2 = k !. The measure p(k) also satisfies (2.6) and has the form

k

p(k)= ~ Wk, j~(X__Yk, j) ' (2.14) j = l

where Yk, 1 < Yk, 2 <" '<Yk, k are the k distinct zeroes of Hk(X ) and

( H txl t 2 , \X_ Yk-~-T] d n (x)/[H~(yk, j) ] . (2.15)

The measure p(k) is also the unique measure supported on k (or fewer) points which satisfies the moment conditions PJ(P)=fii for j = 0 , 1 .... , 2 k - 1 .

For k = 1, 2, 3, one has

p(~ p(2)=�89

+fi)].

122 R.S. Ellis and C.M. N e w m a n

Remarks. 1. The limits (2.10) and (2.12) with k = 2 and p = p(a) were first proved (by ad hoc methods) in [21]. These results are now standard tools in constructive quantum field theory [12]. The results of [21] were extended in [5] to a class of measures more restricted than in the present work and with k = 2; the proofs used facts about large deviations of sums of independent, identically distributed random variables [19] applied to the representation (2.3).

2. We write down measures p, which give rise to a certain subset of the limits in (2.12). Suppose that p is of type k and strength ). and

~exp(X~Z,+elxlk]dp(x)<oo for some e>0. (2.16) \ ~ /

Given real numbers cq, .-.,ak and n~{1,2, ...} sufficiently large, we define

k [- ] dp,=k~exp ~ ajHi(x)/(n!-J/2kj!) dp(x), (2.17) j = l

where k~ is a normalization. Because of (2.16), (2.1) is satisfied for p~ with n sufficiently large. Then p,---, p and

- - 2 t ~ j ! k! j=a J! ] '

where C ~(k!)-1 ~ H~ dp. To prove (2.18), one verifies that the measures p, satisfy the conditions (4.9) which figure in the proof of Theorem 2.2.

We spend the rest of this section outlining how our results extend to the case of spherically symmetric probability measures on ]R t, t e {2, 3,...}. Let gt denote the set of such measures which also satisfy the bound

]p,t

We denote by {-~n)(p):j=l, ..., n} ( n = l , 2 . . . . ) the array of dependent random vectors, taking values in JR*, with joint distribution

1 - - exp [I dp (2j), (2.20) Z, 2n --! j= 1

where Z, is a normalization factor. We denote by N(2) a normal distribution on IR' with mean 13 and covariance matrix I and define moments (j = 0, 1, 2,...)

IR*

l l:JdN,( ) = IR t

1, for j = 0 ,

( 2 j + t - 2 ) ( 2 j + t - 4 ) . . . (t +4)(t +2) t,

for j = 1 , 2 . . . . .


We now assume that p satisfies the analogue of (2.6):

[g22 ] _ j" exp(g dNt(2), ~ exp (g .2 )dp (2 )<eXp ( z ! ~ .2)

for all ~ IW, g+0. (2.21)

Because of the spherical symmetry of p, the function g ~ ~ exp(~. 2)dp(2) is ]R t

actually a function of Igl2. Hence there exists k = k(p) ~ { 1, 2, . . . } and 2 t = 2 t(p) s(0, oe) such that

0, for j = 0, 1, ..., k - 1 f i 2 j ' t - -#2J (P)= ] k t > 0 , for j = k . (2.22)

Theorems 2.1 and 2.2 go over to the present context with natural modifications (e.g., the constant (2k)! in (2.10) is changed). We next give the extension of (par t of) Theorem 2.3. We denote by 6 ( [2J - 7), 7 >0 , the uniform measure, with total mass one, on the spherical shell 121 =7 in IR t.

Theorem 2.4. Given re{2, 3,. . .} and ke{1, 2, . . .} there exists a probability measure p}k) on ]R t of the form

[k+l] 2

p}k)(2)= ~ Wk, j, t6(12l--yk,~,t) (2.23) y=l

such that (2.21) and (2.22) are valid. For all j = 1, ..., [(k + 1)/2], Wk,;, t > 0; if k is even, then O < Yk, l,t < ... < Yk, k/2,~, and if k is odd, then O= Yk, l,t < Yk, 2,t < ,.. < Ya,(k + a )/2,r Thus, p~k) is supported on k/2 spherical shells, with the origin counting as a half-shell for k odd. I t is the unique measure in ~ which is supported on k/2 spherical shells and which satisfies (2.21) and the moment conditions t~2j(P)= fi2j, t for j = O, 1, . . . , k - 1.

For k = 1, 2, 3, we compute

p}~)=6(12l), p}z)=6(12l-1/ t ) ,

p ( 3 ) - _ 2 6(121) + t +-t5 6(12l - 1 /~5 ) . t t + 2

In analogy with Theorem 2.3, it is possible to evaluate

2k, t = sup {2t(p): p ~gt and k(p) = k}, (2.24)

where 2t(p ) is the number appearing in (2.22); to prove that 2k, t = 2t(p} k)) and that p}k) is the unique measure in gt (of type k) with this proper ty ; and to relate the weights Wk, j, t and the mass points Yk,j,t to certain classical or thogonal polynomials (generalized Laguerre polynomials). We discuss these matters briefly at the end of Section IV.

Our results do not seem to extend easily to the case of general measures on IR * which are not spherically symmetric. (For t = 1, this is carried out in the next section.) There is one special case worth mentioning. Let p be a measure on IW

124 R.S. Ellis and C.M. Newmal

which satisfies (2.21) and assume that there exists an integer ke{1,2,. . .} and a measure/5~g t such that

S )~1,..~it dp --~- ~ ~)(~i11,., Xl tdp w h e n e v e r IR t ~t

i~,...,ite{0, 1 . . . . . 2k}, ia+. . .+ir

If/5 satisfies (2.22) for some 2 > 0 (and the same k), then the t-dimensional analogue of Theorem 2.1 is valid for p.

III. General Statements of Limit Results and Their Proofs

Throughout this section, p or p, will denote a nondegenerate probability measur~ on IR satisfying (2.1). X} " ) - X}")(p) and S, =-S,(p) are defined as in Section II. Th( proofs of our theorems depend on four lemmas, which are presented next. Our firs lemma introduces a function G(s)=-Gp(s), which will play a basic role ir determining the asymptotic behavior of S,(p). The importance of G is shown b5 Lemma3.3, which relates the distribution of S,(p) to this function. After th~ lemmas, we give some nomenclature which will be used in the statements of th( limit results, Theorems 3.8-3.9 and Corollary 3.10. The section ends with proofs o these results together with some remarks. For the rest of this section, we write G fol G o and G, for Gp~, where (p,: n = 1, 2, ...} is a sequence of probability measures. All of the work in this section extends, with minor modifications, to the case o spherically symmetric measures on IR', t~ {2, 3 . . . . }.

Lemma 3.1. Given p and real s, we define

S 2 G (s) = ~ - - In ~ exp (s x) d p (x). (3. !

Then G is real analytic and G(s) -* + oo as ]sl -* or. Thus, G has only a finite number oj global minima. Also,

~ e x p ( - n G ( s ) ) d s < o o for any he{l ,2 , ...}. (3.2}

Proof. For complex z and L > 0, we have the estimate

I~ exp(z x) dp(x)l [IZI 2 X2\

<=p([ -L ,L] )exp(L [zl)+ ~ exp I ~ - + ~ - ) dp(x). (3.31 Ixl>L

Because of (2. I) this has order o(exp([z[2/2)) as l zl ~ o% and so G is real analytic anc tends to +oo as Is]~oe. We next prove (3.2) for n = l :

e- ms)ds = f~ e-(s-x)2/2 e~2/2 dp (x) ds = 1 ~ ~ e ~/2 dp (x) < oo. (3.41

For n>2 , (3.2) follows from (3.4) and the lower boundedness of G. []


Lemma 2. Suppose that for each n, W, and I1, are independent random variables such that W, ~ v, where

S elaX dv(x).t:O, all aEIR.

Then I7, ~ I~ if and only if W, + I1, --, v �9 t~.

Proof Weak convergence of measures is equivalent to pointwise convergence of characteristic functions. []

Lemma 3.3. Suppose W ~ N (O, 1) is independent of S , = S,(p) for all n > 1. Then given 7 and m real,

W S , - n m exp(-nG(s /n~+m)ds nl/2_~ + - - (3.5) nl-~ Sexp(-nG(s/n~+m))ds"

Proof Given 0 real, we let I denote the interval ( - o% nl-~O+nm]. We have by (2.3)

f W S , - n m O t = p { I / f s EI} P lnl/2_7+--n~_ ~ <=

J

1 = Z~- ! d IN(0, n) * {exp (x2/2 n) p*"}]

I (3.6)

--l~(nl-2']~Z,\~/_~ i e x p ( - n G ( n T + m ) ) d s ' s

Taking 0-~ oo gives an equation for Z, which when substituted back yields (3.5). The integral in (3.5) is finite by (3.2). []

Remark 3.4. For 7 <�89 the random variable W does not contribute to the limit of (3.5) as n -~ oo. For 7 =�89 the presence of W in (3.5) leads to the extra term of - 1 in (2.11), (3.23), and Corollary 3.10.

Lemma 3.5. Suppose that p, are measures which satisfy

exp (X2/2) dp, ~ e x p (x2/2) dp. (3.7)

We define

f = rain {G(s) lsEl(} (3.8)

and let V be any closed (possibly unbounded) subset of ]R which contains no global minima of G. Then there exists e > 0 so that

e "I ~ e-"~'(s) ds = O(e-"~), n --* oe. (3.9) V


Now assume that for some m real and some positive integer k, G~ j = 1, 2, . . . , 2 k, satisfy

G])(m) = 2 jn 1 -j/2k + 0 (1/n 1 - j/2k), n ~ o% (3.10)

where 21, ..., 2ak are some real numbers with 2Zk>0. Define

B ,(s ; m) = G, (s + m) - G n(m ). (3.11)

Then there exists 6 > 0 sufficiently small so that as n ~ oc

2 k

nB,(s/nl/2k; m)= ~ 2jsJ/j! ~= 1 (3.12) {isl2k+l + 0 \ n-~f~u]+o(1)Pl(s ) for Isl<~n 1/2k,

and

1 2Zk s2k+P2(s), for [s l<fn 1/2g, (3.13) nB,(s/nl/2k; m) >~ (2 k)!

where P1 is a polynominal of degree 2 k and Pz is a polynominal of degree 2 k - 1.

Remark 3.6. We use (3.10) in the proof of Theorem 3.3., which in turn will yield Theo rem 2.2 after some addit ional work in Section IV.

Proof. By (3.3), applied to the p, 's , and by (3.7), we see that

S exp (z x) dp,(x) ~ exp (z x) dp(x), uniformly on compacta of C,

and that

~exp(zx)dp,(x)=o(exp(]z[2/2)) , ] z [~oo , uniformly in n.

Hence

G~)~G u) uniformly on compac ta of 1R, for any je{0 , 1, 2, ...}, (3.14)

and

lim {infG,(s)} = + oe. (3.15) isl~oo n

(Compar ing (3.10) and (3.14), we see that ~2k = G(Zk)(m) .) Facts (3.14)-(3.15) imply that there exists e > 0 so that for n sufficiently large,

inf {G,(s)[se V} > inf {G(s)[selR} + e = f + ~.

Hence

enf S e-na"(S) ds <enf e-(n-1)(f +~) ~ e-6"(S) ds V V

< e "I e -"(f+~) [e (s+~) ] ~ ~ exp (X2/2) dp,(x)] = O(e-"~), n--+ o%


which is (3.9). Fact (3.14) implies that there exists 51 >0 so that

sup B,(s; ~G~J)(m)S; m ) - . -=O(IsJZk+l), for IS1<81. j = l

Equality (3.12) follows from (3.16) and (3.10). Now

s 2k I Gy)(m)f( B.(s; m)>=GTk)(m) ( ~ . - B,(s; m)--)= l ~' " sJ

_ " S j

while

(3.16)

(3.17)

2k+l ~ 1 2k T O([sl )=g22kS /(2k)., for Is[<6 a,

for sufficiently small 82. (3.13) now follows from (3.10), (3.16) and (3.17) with 8 = rain (61, 82).

Nomenclature. Given a positive integer ~, e distinct real numbers ml, ..., ms, and c~ positive integers k 1, ..., ks, we say that the vector (ml, kl; ... ; m~, k~) is admissible. We write

p ~ ( m l , k l ; . . . ; m~, k~)

if the set of global minima of G is {m 1 . . . . , m~} and for each i= 1, ...,

G(s)=G(mi)+ 2i (s-mi)2k' f-o((s-mi) 2k') as s--*mi, (3.18) (2 k,)!

where 2 i is a positive real number. We call k(ml) -k i the type and 2(mi) -2 i the strength of the minimum m i. (These definitions also make sense for a local minimum; see Remark 3.12.) The maximal type is defined as the largest of the k~'s. The measure p is said to be pure if G has a unique global minimum and semipure if it has a unique global minimum of maximal type. A pure (or semipure) measure is said to be centered at m, the location of the unique global minimum (of maximal type); its type and strength are k(m) and 2(m), respectively.

Remark 3.7. In Section IV we connect this nomenclature with the assumptions made in Section II. Meanwhile, it suffices to note that (2.6) is equivalent to the assumption that p is pure and centered at 0.

We next state our main results which extend the theorems of Section II. See the remarks at the end of this section for further results obtained by means of conditioning and for the existence of measures satisfying the hypotheses of our theorems.

Theorem 3.8. Assume that p~(ml , kl; ... ; m~, k~), where the latter is assumed admissible. Then

S, ! ~ bi3(s-ml), (3.19)

Yl i=1

128 R.S. Ellis and C.M. Newmar

where

bi=S[2(mi) ] . -1/2ki, tf k i is maximal. (3.20)

[0, otherwise.

Theorem 3.9. Suppose that {p,: n = 1, 2, ...} are measures satisfying (3.7). Suppose further that p is pure, centered at m, of type k and strength 2zk, and that

G~)(m)_n 2 J ( 1 ) -j/2k ~-0 ~ , j = l , 2 , . . . , 2k - -1 , n ~ c ~ , (3.21)

for some real numbers 21 . . . . ,22k-1' Then

s.(;.) - - - ~ fi(s - m) (3.22)

n

and

S , ( p , ) - n m I N ( - 2 1 / 2 2 , 2 ~ - I ) , /f k = l , H1 -- 1/2k -- '+] / 2k X (3.23)

~.exp { - ~ 2jsJ/j!], /f k>2, \ j = l /

where 221 _ 1 > 0 for k = 1.

Corollary 3.10. Suppose that p is pure, centered at m, of type k, and strength )~. Then

S, - n m ~'N(0, )~- ~ - 1), /f k = 1, n 1-~/2k ~ ) exp(-2s2U/(2k)!), if k>2,

where 2-1 _ 1 > 0 for k = 1.

We prove Theorems 3.8 and 3.9. If the latter we set p, = p for n = 1, 2, ..., )~ . . . . = "~2k- t ----- 0, 22k = 2, then Corollary 3.10 follows. The main idea of the proofs is that because of(3.5), the contribution of G to the limits is due only to the global minima of G of maximal type.

Proof of Theorem 3.8. By Lemmas 3.2 and 3.3 (with 7 -- 0), it sufficies to prove that for any bounded continuous function h

Se-"G~)h(s)ds ~ ~ h(m~)b~ b~. (3.24) e-nG(s) ds i= t

We work with the number ~ from (3.12)-(3.13), decreasing it (if necessary) to assure that O < 6 < m i n {[m~-mjl: 1 <i:~j<~}. We denote by V the closed set

V = IR- ? (m~-6, m~ +6).

By (3.9) there exists ~ > 0 so that

e"r S e-"a(S)h(s)ds=O(e-"~), n ~ oo. (3.25) V


For each i = 1, . . . , e, we let k=k(mi) , 2=2(mi). Then

mi+O

nl/Zk e "y ~. e-"O(S) h(s) ds m l - 6

6

= n 1/2k ~ exp [ - n(G(s + ml) - G(mi))] h(s + mi) ds -a (3.26) (s )

= f exp [--nB(s/nl /2k; mi)] h n~T~+rn i ds Isl < 6n 1/2k

=h(rni)2-1/EkSe-S'~/(2k)! ds+o(1), n ~ o e .

In deriving the last equation, we used (3.12) applied to

0, for j = l , . . . , 2 k - i , G~)(mi) = 2~ = G(J)(ml) = 2, for j = 2 k.

We also used the dominated convergence theorem, which is valid because of(3.13). Now (3.24) follows from (3.25) and (3.26) applied separately to the numera to r and denomina tor on the left-hand side of (3.24). [ ]

Proof of Theorem 3.9. We first verify that (3.10) is valid for j = 1, . . . , 2 k. This is of course the same as (3.21) for j < 2 k. By (3.18) and the hypotheses on p, we see that G(2k)(m)=22k; since the p, 's satisfy (3.7), (3.14) the implies that (3.10) holds for j = 2 k .

Concerning (3.22), it suffices to prove that

exp( - n G,(s)) h(s) ds -~ h(m) (3.27)

.[ exp ( - n G ,(s)) ds

for any bounded continuous function h. Let V = { s l [ s - m ] > 6 }, where c5 is the number in (3.12)-(3.13). By (3.9), there exists e > 0 so that

e"Z [ e-"~"(S)h(s)ds=O(e-"~), n ~ o e . (3.28) V

Since G,(m) ~ G(m) = f , we have

m+6 6 nl/Zke "z [. e-"G"(s)h(s)ds=nl/Zke"(") ~ e- ," , (s ,m)h(s+m)ds

" -~ -~ (3.29) 2k

with q(n)=-n [ f - G , ( m ) ] = o (n). In the last equality in (3.29), we used (3.12)-(3.13) and dominated convergence. The limit (3.27) follows from (3.28)-(3.29).

Concerning (3.23), it suffices (by Lemmas 3.2 and 3.3 with 7 = 1/2k) to prove that

S exp [ - n Gn(s/n 1/2k + m)] h(s) ds

exp [ - n G,(s/n 1/2k + m)] ds (3.30)

--~exp (--j~l ~ISJ/'') h(s)ds/j'exp (-j~l~jsJ/'!)'s


for any bounded continuous function h. We pick 6 >0 as in (3.12)-(3.13). By (3.9), there exists 8 > 0 so that

e "f ~ exp[--nG,(s /n l /Zk+m)] h(s)ds=O(nl/2ke-"~), Isl>__OnX/Zk

By (3.12)-(3.13) and dominated convergence, we have that

n--* oo. (3.31)

e"f S exp [ - n G,(s/n 1/2 k + m)] h(s) ds ISI <~n l /2k

=eq(")[ S exp[--nB,(s/nl /Zk; m)] h(s)ds] (3.32) Isl < ,~n ~/2k

2k

with q(n) = n [ f - Gn(m)] --= o (n). Now (3.31)-(3.32) yield (3.30). To prove 221 _ 1 > 0, we set ~o(s) = ~ exp(sx) dp(x) and calculate

G " ( s ) = I - ~ Ix ~ x e x p ( s x ) d p ( x ) ] 2exp(sx) dp(x) q~(s) ~o(s)

=- 1 - O ( s ) ,

where O(s) > 0 since p is nondegenerate. If p has a type 1 minimum at m of strength 2z, then ) :2=G"(m)>0 and so

2y t _ 1 = O(m) > 0 [~ 1~2

Remark 3.11. The result, in Theorem 3.9, that ,l 2 1 1>0, can also be derived from inequality (2.13) of Theorem 2.3 (which is shown in Section IV to apply to the strength of any pure p of type k) together with the fact that p is nondegenerate.

Remark 3.12. Corollary 3.10 may be extended by means of conditioning. Let p be arbitrary and let m be any type k local minimum of G of strength 2. Then one may choose e >0 sufficiently small so that the conditional distribution of (S, - n m ) / n 1-1/2k, given that [(S, /n)-m[<e, tends to the right hand side of (2.10). Theorem 3.9 may be similarly extended. Also, by this technique, Corollary 3.10 as stated may be extended to the case of semipure measures. One combines the conditioned limit result with Theorem 3.8, which implies that a weak law of large numbers is valid if (and only if) p is semipure; thus, for p semipure, P{I(S,/n) -m] <8}--, 1 for all 8> 0, where m is the centering. Proofs of these and related results will appear elsewhere.

Remark 3.t3. As an extension of Theorem 2.3, it can be shown that given any admissible vector (m 1, kl ; ... ; ms, ks), there exists a unique probability measure 13 supported on k 1 + " " + k~ points such than f3 ~ (ml, k 1 ; ... ; ms, ks). Furthermore, if p :~ fi and also p ~ (m 1, kt; ... ; m~, k~), then p is not supported on fewer points than


is p, and 2o(mg) < 2~(mi), each i= 1, ..., ~. The proof of these facts, which will appear elsewhere, uses techniques from the theory of moment spaces [16]. Moment space methods are also used heavily in Section IV (see the proof of Theorem 2.2 and the ciscussion which follows that proof).

IV. Proofs of Theorems of Section II

We first prove several lemmas which relate the hypotheses of Theorems 3.8-3.9 and Corollary 3.10 to the moments ofp and p. and then prove Theorems 2.1-2.2. This section ends with proofs of Theorems 2.3-2.4.

We define the moments I~j=#j(p) and fij by (2.4)-(2.5) and the cumulants Kj =Kj(p) by

exp(zx) dp(x) = exp KjzJ/j! . (4.1) j=O

This series converges in a neighborhood of z=0. It follows from (4.1) that for a probability measure p, g I =]21 and for j > 2

g j -- #j = 2 c ] ( m l , ' " , m j_ 1) (11) ml... (K j_ 1) my-1 (4.2)

-= Z dj(ml, "", mj_ 1)(#1) . . . . . (#j_ 1)mY-1.

Both sums in (4.2) are over only those integers m~ => 0 satisfying

j - 1 Imz= j (4.3)

/ = l

and the coefficients cj and dj are independent of p. We also use the following facts about the Hermite polynomials {Hi(x): i=0, 1, ...}:

0, i ~ j, (4.4) ~Hi(x)Hj(x)dn(O , 1)(x)= i!, i=j ,

and

i-1 Hi(x)=xi + ~ qi, j x J - x i +Qi(x), (4.5)

j = o

where Qi is a polynomial of degree less than i.

Lemma 4.1. Let p be as in Section III. Then G has a global minimum at the origin if and only if

5exp(sx)dp(x)<=exp(sZ/2), all s real.

This is the unique global minimum if and only if the inequality is strict for s 4= O; i.e., if and only if (2.6) holds. G has local minimum at the origin of type k and strength 2 if and only if(2.7) is valid. Thus, a measure p is pure, centered at the origin, of type k and strength 2 if and only if (2.6)-(2.7) hold.

132 R.S. E l l i s a n d C . M . N e w m a n

Proof The first two statements are immediate consequences of the definition of G. By (4.1) G has an expansion near the origin given by

Gp(s)= - K ~ s+(1 - K 2 ) s2/2 - ~ KjsJ/j!. (4.6) j = 3

Thus a (local) minimum of type k and strength 2 occurs at the origin if and only if K 2

= 1, K 1 = K3 = K 4 . . . . . K_2k -- 2 = K z k - 1 = O , K 2 k = - - •. Since the cumulants / ( j of N(0, 1) vanish except for K 2 = 1, the third statement in the lemma follows from (4.2). []

We next turn to the relevant moment conditions on p, which are equivalent to (3.21). We claim that without loss of generality it sufficies to do this for m = 0. Indeed if m + 0, then define the measure

dis (x) = exp (m x) dp (x + m)/~ exp (m x) dp (x + m). (4.7)

Since

G~ (s) = G o(s + m) - G p(m), (4.8)

the existence and nature of a minimum of G o at m is equivalent to the corresponding facts about G a at the origin.

Lemma 4.2. Suppose that p, , P2, " " are probability measures such that G,(s)=S2/2 - l n 5 exp(sx) dp,(x) exists for n = 1, 2, ... (at least for small s). Then (3.21) is valid with m = 0 if and only if

#J(P,) = flit- 2 jn l -j/2k + 0 (1/n 1 -j/2k), j = 1, 2, ..., 2 k - 1. (4.9)

Proof By (4.6), (3.21) is equivalent to

Kj (p,) = K ~ - 2 in I - j/2 k "4- 0 (1/n 1 - j/2 k), (4.10)

j = 1 , 2 , . . . , 2 k - 1 , as n ~ o o .

An application of (4.2) then shows that (4.10) is equivalent to (4.9). []

Proof of Theorem 2.1. Lemma 4.1 implies that a measure p is pure, centered at the origin, of type k and strength ,% if and only if (2.6)-(2.7) hold. Hence (2.9)-(2.11) follow from Theorem 3.8 and Corollary 3.10. []

We next prove Theorem 2.2. In the proof, we appeal to Theorem 2.3 which is proved later in this section independently of the present result.

Proof of Theorem 2.2. By Theorem 3.9 and Lemma 4.2, it suffices to prove the existence of measures p, satisfying (3.7) and (4.9). Pick real numbers %--+0 as n ~ oo such that

1 1 1 �9 0 as n--+ oo. (4.11)

an n 1-(2k-1)/2k O~nnl/2k

It suffices to choose probability measures {v,: n = 1, 2, ...} such that

sup ~ ex2/Zdv,(x) < oo (4.12) n


and for large n,

pj(v,)=fij-2/a,n l-j/2k, j = 1, 2, . . . , 2 k - 1. (4.13)

For then, the measures p, = ( 1 - a , )p + a, v, satisfy (3.7) and (4.9). Given c > 0, we define a subset I c of ]R 2k - 1 :

I c= {fi=(#~, . . . , #2k-1): for some probabil i ty measure v (4.14)

with support in [ - c, cJ, #~ =#j(v), j = 1 . . . . , 2 k - 1}.

Ic is closed; its interior I ~ is a nonempty open connected set [16; Ch. 2]. It suffices to prove that

--(ill . . . . , fi2k- l) ~I~ for some ce(0, ~ ) , (4.15)

for then we can find measures v with support in [ - c , c] such that the vectors fi(v)-= (/~1 (v) . . . . ,/Z2k_ I(V)) cover some small ne ighborhood of ~. Thus by (4.11), for large n, measures v, can be picked to satisfy (4.12) and (4.13). To prove (4.15) we appeal to Theorem 2.3 which guarantees the existence of a measure p(k) with support on k points such that [z(p ~k)) = ~. We now pick ce(0, oe) so that the support o fp (k) is contained in ( - c, c); hence ~elc . Since, according to Theorem 2.3, the only probabil i ty measure v supported on k or fewer points with/~(v) = ~ is v = p(k), it follows by [16; Ch. 2, Thm. 2.1] that /~ is not a boundary point of I~. [ ]

We now turn to the proofs of Theorems 2.3 and 2.4, for which there are two related methods of attack: the theory of Gaussian quadra ture and the theory of momen t spaces. While we could have used the former to prove both theorems, we base the proof of Theorem 2.4 upon the latter, which is the more efficient of the two in treating existence of extremal measures. After the p roof of Theorem 2.4, we note what informat ion the theory of Gaussian quadra ture provides. In this paper the theory of momen t spaces is also used in the proofs of Theorem 2.2 and the result stated in Remark 3.13.

Proof of Theorem 2.3. Let f be any function in Czk(1R) which has polynomial growth at infinity. Define p = p(~) by (2.14). Using [11; w we have the Gauss- Hermi te quadra ture formula

d 2k

f f dN=S f dP+ 5 ~b2k d~T~ f dx, (4.16)

where 1~)2k is an even, strictly positive function on IR. Setting f=xJ, j=O, 1, ..., 2 k . . . . . we see that

[=f i j , j = 0 , 1, . . . , 2 k - 1,

/~j(p) J=f i j = 0, odd j > 2 k + l

( < p j, even j > 2 k.

(4.17)

134 R.S. Ellis a n d C.M. N e w m a n

Thus, p satisfies (2.6)-(2.7). We now prove that 2(p)=k!:

~ - ~,~(p) = j x ~ d ( N - p) = S ( U ~ - y~)~ d ( N - p)

= ~ H 2 d(N - p) - 2 ~ H k Qk d(N - p) + ~ Q.~ d(N - p) (4.18) 2 2 =~Hk dN-[Hk clp=k!.

The second equality uses (4.5), the fourth the first line of (4.17) (H k Qk and Q~ are polynomials of degree 2 k - 1 and 2 k - 2 , respectively, and so each has the same dN and dp integral), the fifth (4.4) together with the fact that p is supported on the zeroes of H k. We now prove that if o- ~ p(k) is any other measure satisfying (2.7), then

2(a) - ~2k-- #Zk(~) < k !. (4.19)

Using the same steps as in (4.18), we find

~,~(o)- ~(p(k~) = ~ u ~ d~ > o.

Hence,

2(if) < fl2k - U2k(P (k)) = k!,

which is (4.19). For any )~(0, k!), any a t [0 , 1], and any distinct integers i, j >= 1, the measure

2 k [1 )~ )+ ~ -- ~. ]1 ( a p(k + i) + (1-- a) P (k + j))

0, j = 0 , 1 . . . . , 2 k - l ,

f i j- /~j(z)= 0, odd j > 2 k + l , 2, j = 2 k ,

t a j > 0 , j = 2 k + 2 , 2 k + 4 , ....

Consequently, z satisfies (2.1) and (2.6) and is of type k and strength 2, as desired. The last statement in the theorem is proved exactly like [23; Thms. 3-4; pp. 137- 1391. []

Remark 4.3. We comment on the existence of quadrature formulae for an arbitrary probability measure a. For convenience, we assume that the support of a is a closed interval I. Given a positive integer k, we seek a probability measure p with support on k points contained in I and a function (~2k which is strictly positive in the interior of I such that for any function f s cZk(/)

d 2k

~ f d a = ~ f d p + ~ q ~ 2 k ~ x z k f dx. (4.20) I I I

In [23; w w it is proved, under the hypothesis that I is bounded, that a unique p and 4~ 2k exist so that (4.20) is valid. In [ 11 ; w 2:3, Ch. 4], (4.20) is extended to

satisfies


certain cases where I is unbounded. Besides the Gauss -Hermi t e case, which we have just treated, this includes the measures c e x p ( - x ) x ~ dx on [0, oo), where cr > - 1 and c is a normal iza t ion constant [11; w We discuss these measures after the p roof of T h e o r e m 2.4, which concerns spherically symmetr ic vector Curie-Weiss models.

Proof o f Theorem 2.4. I t suffices to consider the corresponding radial measures on [0, oo). Thus we seek a measure a(r) on [0, or) such that

co

Sr2JdG(r)=yr2;d]Nt](r) , j = 0 , 1, . . . , k - 1, (4.21) 0 0

where

co

d IN, I (r) = r ' -* e - ~/2 dr/~ r ' - ~ e -r~/2 dr 0

is the radial measure corresponding to N,(2). For j e {0, 1,... }, define uj(r) = r2J; then the sets {u0,ul, . . . ,Uk_l}, and {u o, ul, ...,Uk} are Cebygev systems on [0, oo) [16; p. 9]. Hence, by [16; Ch. 5], there exists a measure a with suppor t on k/2 points (with 0 count ing as a "ha l f -poin t" for k odd) such that (4.21) holds; ~ is the minimal ly suppor ted measure on [0, 0o) which satisfies (4.21). In addition, one has (analogously to the last line of (4.17)) that

co co

S r21da(r) < ~ r2JdJNt](r), j = k , k + 1, . . . . (4.22) 0 0

We let p}k) be the measure in ~ whose radial measure is a and note that (2.21) (2.22) follow f rom (4.21)-(4.22). [ ]

The theory of Gauss ian quadra tu re does not apply directly to (4.26) because there are no condit ions on the odd moments . However , by mak ing the change of variables x = r 2 and considering separately the cases k even and k odd, one can conver t (4.26) to a form for which Gauss ian quadra tu re methods work. We omit details (see [24; w but summar ize some of the results. Fo r ae(1, oo), we define po lynomia l s

1 d zi . I3Y} (x )=- -x -~eX~(x~+'e -X) , i = 0 , 1, ..

' " " i ! dx ~ "'

known as the generalized Laguerre polynomials [11;w w They form an o r thogona l sequence on [0, oo) with respect to the measure e-" x ~ dx. We denote by 0 < y},~)~ < . . . < Y~,~I the i distinct zeroes of/3~')(x). One can prove the following relat ion between the mass points Yk, j,t of the measures p}k) in Theo rem 2.4 and the zeroes of /31):

Yk, j,t=tz-.yw , j = l , . . . , ~ , for k even,

k - 1 ~2- ~+ 1) ~ yk, a + l , t = t Yk2~ / , j = l , . , for k odd,

- - ' - ' 2 '


where c~ = t/2 - 1 (recall Yk, *,t = 0 for k odd). The weights %, j,t of pl k) as well as the number 2k, t in (2.24) may also be computed in terms of the D~ ).

V. Physical Content of Limit Results

We define a general class of Ising spin systems and then specialize to the case of Curie-Weiss models. For each A cTI d, there is a family of t-dimensional random vectors {2~: jeA}. J~r represents the spin, or magnetic moment, of the individual atom at the location j in a magnetic crystal whose macroscopic shape is described by A (so that in many physical situations one has t -- d = 3). When the crystal is kept at a fixed temperature, the joint distribution is given by

1 ~A exp(--flHA({2~})) [I dp(2j), (5.1)

jEA

where fl > 0 is proportional to the inverse temperature, H A is the energy (a function on (IR~) A, the space of configurations of the spins), p is the probability distribution of a single spin in the absence of interactions with the other spins (or equivalently in the limit ft-+0), and 2~ a is a normalization constant. We are interested in ferromagnetic systems with only isotropic pair interactions and with no external magnetic field. For such systems H a has the form

HA = _ 71 ~ Jija 2i ' 2j (5.2) z,J

with Ji~ > 0; also p is assumed to satisfy (2.19) so that (5.1) exists as a finite measure, possibly only for small ft.

In physically realistic models one typically takes J[)=J(lli-j[I), where [I'Ll denotes Euclidean distance in Z a and J is some nonnegative function on [0, ~ ) which is rapidly decreasing at ~ . Because of the latter condition, we speak of the interaction as being short-range. In order to obtain a stationary family, {)~j: j~TZfl}, and in order to rigorously investigate such phenomena as phase transitions and critical points, one considers the limit, in an appropriate weak sense on ( R t y , of (5.1) as A--+;g a.

In a Curie-Weiss model one takes Ji~=J/iA[ for all i,jeA, where [.I denotes cardinality and J is a fixed positive number; thus each spin interacts equally with all others. Such a model can be thought of as approximating the above short-range

interaction models if we set J = ~ J(2P)d~p. If we replace J(lli-jl[) by ~a J(7 l[i-jql), IRa

> 0, let A --+ 7/d and then let ~ --+ 0 + (the limit of a weak long-range interaction), we obtain the same thermodynamic behavior as exhibited in the A -+ 2U limit of the mean field model with this value of J [26; p. 105]. Without loss of generality, we may simplify the Curie-Weiss model by taking A = {1 . . . . . n}, I11 = n , J = 1, and by scaling the random vectors by l/ft. We then obtain (2.20) (or (2.2) for t = 1) with dp replaced by

dp~ (~) -- dp (~/1/~). (5.3)


For the remainder of this section we restrict attention to this model with t = 1 and assume that (2.1) is valid with p replaced by p~ for all fi under consideration.

In the Introduction it was stated that typically an Ising spin system has a critical value fic of/?: for fie(0, fie), the central limit theorem for S, is valid while for f i= fic a limit as in (1.1) holds for some v~(1,2). (Below we discuss what happens for fie(fie, oo).) Appealing to Theorem 2.1, we now claim that ifp satisfies (2.6) and is of type k with k > 2, then we have fic = 1 for the corresponding Curie-Weiss model. The second line of (2.10) shows that for f i = l (1.1) is valid with v= 2k/(2k-1) . Hence it suffices to show that for fie(0, 1) p~ satisfies (2.6) and has type 1. We have

~exp(sx)dp~(x)=~exp(sx]/fl)dp(x)=e~S2/2<e s2/2 for s real, s+0 ,

and

#2(pr = f i < 1.

Thus the claim is proved. Another way of stating the properties ofp~ which will be of use below is that GoB has a unique global minimum at s = 0 which is of type k > 2 for f i= 1 and of type 1 for fie(0, 1).

As to the status of limit theorems for Curie-Weiss models with fie(fir oo), we single out a class of measures p for which S,(p) has a non-Gaussian limit if anti only i f / /= 1; for fiG(l, o0) as well as for fie(O, 1) a Gaussian limit arises. Consider the class of symmetric probability measures p with #2(p )= l such that p satisfies the Griffiths-Hurst-Sherman inequality:

d 3 In {~ exp (s x) dp(x)} ds a ~ 0 for all s > 0. (5.4)

A large class of measures is known to satisfy this inequality, which turns out to be of considerable physical importance [13; 6; 8]. Such a p may be shown to satisfy (2.6) and to be of type k > 2. Hence by the discussion in the previous paragraph, a non- Gaussian limit arises for fi = 1 while a Gaussian limit is obtained for fie(0, 1). On the other hand, one can easily prove that for fiG(l, co), Gp~ has exactly two global minima, each of type 1, at s=+m(fi), some m(fi)>0. One may show that conditional upon the average spin being near one of these two values, say m(fi), (S n -nm(fi))/ l /~ tends to a Gaussian limit as n ~ oo. This behavior for large fi is also typical of general spin systems [10].

The change in the number of global minima of Gp~ from one for fie(0, 1) to two for fie(l, oo) when p is a GHS measure shows up in another context. One can prove that for such a p the weak limit (on lR~176 as n --* 0% of the joint distribution of {X~} has a very different form for fie(0, 1] than it has for fie(l, oo). For fie(0, 1] it is an infinite product measure, each factor being the same symmetric measure on IR 1. For fie(l, oo) the weak limit is a convex combination of two infinite product measures, the individual factors having mean m(fi) or -m(fi) according to which ergodic class they belong to. In this region of strong correlation the spin random variables tend to cluster about the two global minima of Gpe; this clustering property carries over to general Ising systems. For a more complete discussion of this phenomenon, see [7].


The same connect ion holds between the global minima of Gp, and the weak limit of the joint distributions for a n o n - G H S measure p and fie(0, oo). However, for such a measure it is more difficult than in the G H S case to trace the evolution of the global min ima of Gp~ as/3 increases f rom zero. We merely point out that for general measures p other topological properties of Gp, such as the presence of local min ima and points of inflection, have impor tan t physical ramifications.

We conclude by briefly discussing other results in the paper. The joint distr ibution of the (") ' X) s in Curie-Weiss as well as in general Ising models often depend on other physical parameters besides ft. Such parameters may be in t roduced via the p dependence in (2.2). One may then interpret Theorem 2.2, Theorem 3.9, and L e m m a 4.2 as pertaining to Curie-Weiss models parametr ized by a number of variables fil . . . . . flq, qe{2, 3 . . . . }, which have a critical point at say (ill . . . . . . , fiq, c). These models give rise to q critical exponents vt, ..., vq and q scaling constants bl, ..., bq so that partial sums

(n) X) (t~l,c+bl/nl/ ' l , . . . , f iq,c+bq/nl/V,) j = l

have a well-defined asymptot ic behavior as n ~ ~ . The measures p(2) and p(3) obtained in Theorem 2.3 are well-known in the

physical literature [26, p. 105; 2]. In the latter, the measure p(3) was used to analyze the tricritical point of liquid helium. The measures p(2), p(3) can be shown to satisfy the inequality (5.4) and to be, in some sense, on the boundary of discrete measures which satisfy the inequality. We conjecture that each measure p(k), any k e {4, 5,. . . }, also has these properties. Numer ica l results support this conjecture.

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Received May 4, 1977

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