Stochastic Geometry in Statistical Mechanics and Quantum Field Theory

Proceedings of the International Congress of Mathematicians August 16-24, 1983, Warszawa

MICHAEL AIZENMAN*

Stochastic Geometry in Statistical Mechanics and Quantum Field Theory

1. Introduction An early success of statistical mechanics has been the provision of a solid framework which encompasses both the laws of mechanics and the basic principles of thermodynamics. Among the spectacular results of this approach are the explanations, from basic principles, of the various phase transitions which are observed in nature. A not less challenging goal is to reach a complete explanation of the critical behavior of bulk systems. The universality of the critical exponents, which has been observed experimentally, is an intrinsically significant effect. I t both calls for, and ïaiscs,, the possibility of a mathematical elucidation of the subject. Great advances towards the formation of a global picture, and the approximate calculation of critical exponents, have been made by renormalization-group related methods. By their nature, these methods do not •offer exact solutions to any specific statistical-mechanical model. (Although they do produce sharp predictions above, and at, the upper-critical •dimension.) Instead, the analysis incorporates the notion of universality, which from the point of view of physics may indeed be better founded than any given model. However, in view of the mathematical intractability of these treatments, the problem of the critical behavior is still well worth further attention. In recent years progress has been made in the rigorous analysis of some of the most important models of statistical mechanics and quantum field theory. Instrumental for this advance has been the identification

* A. P . Sloan Foundation Fellow. Eesearch supported in p a r t by N S F Grant PHY-830H93. . [1297]

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Sektion 13: M. Aizenman

of some stochastic-geometrical effects which play a key role in their phase transitions, and the corresponding critical behavior. The results for statistical mechanics include the following: (1) Proof that the critical behavior of the "susceptibility", in Isingtype models, stabilizes when the model's dimension reaches the value d = 4. Above that "upper-critical dimension" the corresponding critical exponent, y, takes exactly the value 1 — which is suggested by an extremely simple (mean-field) approximation. (2) A simple explanation of certain aspects of the critical behavior which are manifested in low dimensions. Specifically, "hyperscaling" — which fails for d > é9 was proven to be "universally" valid in d = 2 dimensions. (3) A partial result for percolation models, which lends certain tenuoussupport to the physicists' "educated guess" — that these models are also endowed with an "upper-critical dimension", and that its value isd = 6. I shall briefly describe here the stochastic-geometric aspects of these models and of the above results. The basic phenomenon which is discussed is the existence of the "upper-critical dimensions", above which the critical behavior in statistic mechanical systems is quite simple, and below which it is rather non-trivial. This situation carries somewhat opposite implications for the constructive quantum field theory, where the level of difficulties of the goals is found to be in an inverse relation to those posed by the challenges of statistical mechanics. However, since this is also the subject of the talk of K. Osterwalder, only a few words would be said here on the quantum field theory. The contribution to the Congress Proceedings contains only a brief summary of the talk. A somewhat more detailed report on this subject^ and a more complete reference list, are given in Aizenman (1983). 2. Stochastic geometry We use this term when refering to properties of random geometrical objects. An instructive example is obtained by considering two random lines, in Rd, which are obtained as the trajectories of two independent Brownian motions (Wiener processes). The question of how typical is it for these paths to intersect, has been dealt with in the papers of Kakutani


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.(1944) and Dvoretsky Erdös, and Kakutani (1950). The answer can be summarized as follows. 2.1. Let b1,b2eRd be the sets of sites visited during times te(0, oo) by two independent, d-dimensional, Brownian paths which start ut Xi, x2e Rd. Then THEOREM

Pmb(bxnb2

^ 0) =

0

for d > 4,

1

for d < 4.

(1)

An analogous (but significantly different) statement holds for random walks. 2.2. Let œ19 co2cz Zd be the sets of lattice sites visited by a pair of independent simple random walks, with some specified starting points, at times J e Z + \ { 0 } , Then THEOEEM

Prob(coxnco2 = 0 ) is

> 0

for

d>4t,

= 0

for d < 4.

(2)

{Notice that the LHS in (2) corresponds to 1—LHS of (1).) These results exhibit two striking features. First, Theorem 2.1 may be 1 + 1 . This puzzle is resolved by the cellebrated observation that the Hausdorff dimension of the paths is 2 rather than 1. The other curious effect is that as a result of the discretization t h e behavior at the critical dimension, d = 4, is changed from being highdimensional in (1) to the low-dimensional in (2). This is due to the fact, which is not reflected in (1), that with probability 1 the two paths b19 b2 come arbitrarily close to each other, even in d = 4 dimensions. 3 . Percolation model In statistical mechanics one typically deals not with isolated components, as in the above example, but rather with infinite arrays of such elements. This is the case in the Bernoulli percolation model, which is based on a collection of independent random variables {nb} which are associated with the bonds (i.e. unit segments joining pairs of neighboring sites) of the lattice Zd. The variables take the values 1, 0 with the homogeneous probabilities p, 1—p- Por each given configuration of values of {nb}, the

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Section 13: M. Aizenmaa

lattice is decomposed into connected clusters — by regarding a bond a» connecting if % = 1. Let G (x) be the cluster which contains the site x e Zd, and \G(x)\ the number of its points. The model exhibits a phase transition, when p is varied, associated with the spontaneous formation of" infinite clusters. The transition is. manifested in the behavior of the quantities r(x,y)

=Pvob(yeG(x)),

P„=Frob(\G(0)\

= oo)

and «= =£T(0,X),

(3)

X

which are monotone functions of p (with P^p) = 0 for p small enough and limP^p) = 1, at d^2). At some critical point, pc, n(p) diverges (for d^2).

The expected power behavior

* S \Po—P\+V is the basis for the definition of the critical exponent y, whose values m one of the questions we shall address. When pressed for some quick guess for the critical behavior, one could try a "tree approximation", in which Zd is replaced by a Bethe lattice (Cayley tree) preserving the local number of neighbors. The calculation is then trivial, yielding H

l+P \ziA—-xy-p~ = -

with ,B.L. = (2Ä-1)- 1 ,

(B)

and yB.L.

= l j

independently of $ (!).

(6)

This of course is a very unreliable estimate of the actual values of pc and y. Furthermore, one can prove that pc ^ pfmIj', which leaves the second "prediction" even more suspicious. (Indeed, it is contradicted by numerical results in low dimensions.) Yet interestingly enough, these values do offer general bounds.

Stochastic Geometry in Statistical Mechanics and Quantum Field Theory THEOEEM

fl)

1301

3.1. For a general d, Po>pf-Jj(#)

U

m

>

and (ii)

y > l ( = ^-L-).

(8)

In its weak form, (i) is a commonly made observation which is very elementary; (ii) is more recent (Aizenman and Newman, to appear) although its proof is surprisingly easy. The interesting point for us is that both can be proven using the observation that dn~x\dp is related to the probability that two neighboring sites, 0 and 1, belong to a pair of very large clusters which do not intersect. More specifically: 1— p dn *=Prob(0(O)n0(l) = 0 | G(0)BX, dp

G(l)sy),

(9)

where the probability is conditioned on the event on the right side, and is averaged over (x, y) with the normalized weights T ( 0 , X) T ( 1 , y)\%2. The above representation is quite telling. First, since Prob(— ) < 1, it leads to a bound on \dxrljdp\, which when integrated down from pc, where n~l(pc) = 0, directly leads to (ii). (We skip here some details on the how does one settle the question of the continuity of H" 1 at pc.) Furthermore, (9) demonstrates that in order for' y to take some other value than 1, the probability of mutual avoidance of a pair of "incipient" clusters" should necessarily vanish. As we saw, for random walks, this happens only in the low dimensions d < 4. While random clusters are significantly different from random walks, these considerations suggest that there may be an "upper critical dimension" above which the critical exponent y takes exactly the value which it has in the simple "tree approximation". We shall later mention a criterion which lends some support to the physicists' claim that the upper critical dimension is d = 6. The percolation model was presented first only because its geometric features are plainly manifest. However, the basic approach discussed here was first developed in the analysis of ferromagnetic spin systems, for which the results are more conclusive.

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Section 13: M. Aizenman

4. Ferromagnetic spin systems Statistic-mechanical models of ferromagnetism consist of lattice arrays of variables ("spins") ax, x eZd, with probability measures of the form: Q(da)

=lim1exp/}[

V cTxay + h£ax]}n

Qo(dcrx)IZ(A)9

(10)

where Q0(d) is some non-interacting even single-spin measure, and Z(A) is the normalizing factor. The physical parameters of the model are the temperature — jS*"1, and the applied magnetic field — hß"1. A quantity of special interest is the magnetization, defined as : M(ß9h) =*fa0Q(da).

(11)

For sufficiently large values of ß, M(ß9 h) — as a function of h is discontinuous at h = 0. A critical value of ß is defined by the divergence of the susceptibility: X(ß) =

Sh

(12) A~0

I t is expected that the critical exponent y defined as y

-]jm

7

l gX{ß)

°

ß/,ßelogll(ßc-ß)

(13) K

}

is (along with the other critical exponents) independent of many of the details of the model. The phase transition is a cooperative effect in a system with an infinite number of degrees of freedom. Consequently, %(ß) is not expected to be derivable from any a priori suggestive, finite, system of differential equations. However, a description of this sort is obtained within the very simplistic mean-field approximation, in which M = M(ß9 h) is the solution of: M =f(2dßM + ßh), with fß) =Jae"aeo(äcr)lJeKaQ0(da).

(14)


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For a general class of distributions, which satisfy the Griffiths-HurstSherman inequality (Griffiths, Hurst and Sherman, 1970; Ellis, Monroe and Newman, 1976; Ellis and Newman, 1978), the function/(A) is antisymmetric and concave on [0, oo]. In such cases (14) leads to the following "predictions": (1) (2) For

ß0 - \2d(o*\Y\

where ê)9 is that the probability of two long current clusters, which contain a given pair of sites, to be totally disjoint is uniformly positive. This framework permits to approach the problem in a way which is somewhat analogous to the analysis of the random walk's property (2). Such arguments have led to the following result (Aizenman, 1982 ; Aizenman and Graham, 1983) : 4.1. Ln systems with Q0 in the Griffiths-Simon class (which includes the cp4, and the Ising spins) the function %(ß), for ß < ßc9 satisfies: THEOREM

(2d)-i(ßc^ßr^X(ß)^ with some c, c< oo.

cid^^-ß)-1, cieoHße-ßr'li+iogiKßo-ßil

d>4, d=i


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The upper bounds in (20) are consequences of the geometrically inspired inequality _1^ d% 2d! dß

^[l + (2dß)^^0a^r1

(21)

which shows that a sufficient condition for y to attain its mean-field value is the uniform boundedness (for ß < ße) of :

£=|S 2 2 °) = T^vT J #i---#/ r ^) 3 -

(23)

aî,2/

The proof of Theorem 4.1, for which d = 4 is the critical dimension, is completed by the Gaussian bound of Fröhlich, Simon and Spencer (1976) which implies that for ß < ß0 (and any d) :

P +X In the above analysis, plays a role which is somewhat similar to the hitting probability for random walks. Another proof of (20), for d > 4, has been given in the work of Fröhlich (1982) by means of a random walk expansion. Had the analog of (24) been known for r(p), at least for d > 6, it would have implied that for percolation y = 1 in d > 6 dimension. However, such a bound has not been derived, and is not expected to hold there as a dimension-independent inequality.

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Section 13: M. Aizenman

6. Few comments on the quantum field theory Above the upper-critical dimension not only do the critical exponents simplify, but also the correlation functions acquire a simple form — the random fields defined by their scaling limits are Gaussian. This fact has somewhat opposite implications for the statistic mechanician and the quantum field theorist. The former's task involves finding the critical behavior — which, as explained above, is simple above dc and quite nontrivial in low dimensions. The goals of the latter require setting models of interacting — i.e. non-Gaussian, fields. This goal is made more difficult by the fact that at least the simplest candidate for such a construction fails — due to the attraction to a "Gaussian fixed point". Conversely, below the critical dimension there is a rigorizable perturbation theory which leads to non trivial fields — which presumably also describe the critical regime of statistic mechanical models. Here too, stochastic-geometric represetnations offer frameworks for the proofs of the above assertions — for which an important insight is derived from Theorem 1. Eecent results in this vane can be found in references (Aragao de Oarvalho, Oaraciolo and Frölich, 1983; Aizenman and Graham, 1983; Brydges,-Fröhlich and Sokal, 1983).

References Aizenman M., 1981, Phys. Bev. Lett. 47, p . 1; and 1982, Gommun. Math. Phys. 86, p . 1. Aizenman M., 1983, Eigorous Eesults on the Critical Behavior in Statistical Mechanics. In : J. Fröhlich (ed.), Scaling and Self-Similarity (Benormalization in Statistical Mechanics and Dynamics), Birkhäuser, Boston-Basel-Stuttgart. Aizenman M. and Graham E., 1983, Nucl. Phys. B225 [FS9], p . 261. Aizenman M. and Newman C. M., Tree Diagram Bounds and the Critical Beahvior in Percolation Models, to appear in J. Stat. Phys. Argao de Carvalho C , Caraciolo S. and Fröhlich J., 1983, Nucl. Phys. B215 [FS7], p . 209. Brydges D. C , Fröhlich J . and Sokal A. D., 1983, Gommun. Math. Phys. 91, p . 117. Dvoretsky A., Erdös P . and Kakutani S., 1950, Acta Sei. Math. {Szeged) 12, p . 75. Ellis E. S., Monroe J . L. and ISTewman C. M., 1976, Gommun. Math. Phys. 46, p . 167. Ellis E . S. and Newman C. M., 1978, Trans. Am. Math. Soc. 237, p . 83. Fisher M. E., 1967, Phys. Bev. 162, p . 480. Fröhlich J., Simon B. and Spencer T., 1976, Gommun. Math. Phys. 50, p . 79. Fröhlich J., 1982, Nucl. Phys. B200 [FS4], p . 281. Glimm J. and Jaffe A., 1974, Phys. Bev. DIO, 536. Griffiths E . B., 1967, Gommun. Math. Phys. 6, p . 121.

Stochastic Geometry in Statistical Mechanics and Quantum Field Theory Griffiths E. B., 1969, J. Math. Phys. 10, p. 1559. Griffiths E., Hurst C. and Sherman S., 1970, J. Math. Phys. 11, p. 790. Kakutani S., 1944, Proc. Japan Acad. 20, p. 648. Lebowitz J. L„ 1974, Gommun. Math. Phys, 35, p. 87. Simon B. and Griffiths E., 1973, Gommun. Math. Phys. 33, p. 145. DEPARTMENTS OF MATHEMATICS AND PHYSIOS RUTGERS UNIVERSITY NEW BRUNSWICK, N.J. 08903, USA

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