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This IMA Volume in ~athematics and its Applications PERCOLATION THEORY AND ERGODIC THEORY OF INFINITE PARTICLE SYSTEMS represents the proceedings of a workshop which was an integral part of the 19R4-85 IMA program on STOCHASTIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS We are grateful to the Scientific Committee: naniel Stroock (Chairman) Wendell Fleming Theodore Harris Pierre-Louis Lions Steven Orey George Papanicolaoo for planning and implementing an exciting and stimulating year-long program. We especially thank the Workshop Organizing Committee, Harry Kesten (Chairman), Richard Holley, and Thomas Liggett for organizing a workshop which brought together scientists and mathematicians in a variety of areas for a fruitful exchange of ideas. George R. Sell Hans Weinherger PREFACE Percolation theory and interacting particle systems both have seen an explosive growth in the last decade. These suhfields of probability theory are closely related to statistical mechanics and many of the publications on these suhjects (especially on the former) appear in physics journals, wit~ a great variahility in the level of rigour. There is a certain similarity and overlap hetween the methods used in these two areas and, not surprisingly, they tend to attract the same probabilists. It seemed a good idea to organize a workshop on "Percolation Theory and Ergodic Theory of Infinite Particle Systems" in the framework of the special probahility year at the Institute for Mathematics and its Applications in 1985-86. Such a workshop, dealing largely with rigorous results, was indeed held in February 1986.



Rapid Convergence to Equilibrium of Stochastic Ising Models in the Dobrushin Shlosman Regime

We show that, under the conditions of the Dobrushin Shlosman theorem for uniqueness of the Gibbs state, the reversible stochastic Ising model converges to equilibrium exponentially fast on the L2 space of that Gibbs state. For stochastic Ising models with attractive interactions and under conditions which are somewhat stronger than Dobrushin’s, we prove that the semi-group of the stochastic Ising model converges to equilibrium exponentially fast in the uniform norm. We also give a new, much shorter, proof of a theorem which says that if the semi-group of an attractive spin flip system converges to equilibrium faster than 1/td where d is the dimension of the underlying lattice, then the convergence must be exponentially fast.
M. Aizenman, R. Holley

Uniqueness of the Infinite Cluster and Related Results in Percolation

We present the following results for independent percolation:
continuous differentiability (in the natural parameters) of the free energy function (mean number of clusters per site),
uniqueness of the infinite cluster,
continuity of the connectivity functions. As a corollary of a) and previous results of van den Berg and Keane, there follows
continuity of the percolation density, except possibly at the critical point.
These results are valid for both site and bond percolation on d-dimensional lattices with arbitrary d and for translation-invariant long range bond models (satisfying a natural irreducibility condition).
M. Aizenman, H. Kesten, C. M. Newman

Survival of Cyclical Particle Systems

We consider here a continuous time interacting particle system on Z with possible states 0,1,…,N-1 at each site. We denote by ξt(i) the state taken at site i and time t. Our models, which we call cyclical particle systems, are specified by the following transition rates:
$${\xi _{\rm{t}}}{\rm{(i)}} \to {\xi _{\rm{t}}}({\rm{i}}) + {\rm{l (mod}}{\mkern 1mu} {\rm{N) at rate }}{\mkern 1mu} \lambda ,$$
$$\lambda = \left| {\,\left\{ {\,{\rm{j}} = \pm 1:\,{\xi _{\rm{t}}}({\rm{i + j) = }}{\xi _{\rm{t}}}({\rm{i) + 1}}\,{\mkern 1mu} {\rm{(mod}}\;{\rm{N)}}} \right\}\,} \right|.$$
. Note in particular that i cannot change state until at least one of its immediate neighbors has state ξt(i) + 1. For convenience, one may equip ξ with a space-time percolation substructure. Arrows from i to i+1 and i-1 are each put dow in a Poisson manner with rate 1. A state change is induced at (t, i) by an arrow at time t which enters i from i+1 or i-1, where the state is ξt(i) + 1. In diagram 1 below, numbers at the points of arrows indicate state change in the realization (N=5). Here, we will always assume that ξ0 has product measure with
$${\rm{P(}}{\xi _0}{\rm{(i) = k) = 1/N}}\;{\rm{for}}\;{\rm{k = 0,}}{\mkern 1mu} 1,...,{\mkern 1mu} {\rm{N - 1}}{\rm{.}}$$
Maury Bramson, David Griffeath

Expansions in Statistical Mechanics as Part of the Theory of Partial Differential Equations

We study perturbations of gaussian processes using a partial differential equation in (infinitely) many variables which describes what infinitesimal change in the perturbation compensates an infinitesimal change in the covariance. We derive a series representation for the solution by iterating an integral equation form of the flow equation and show that the series is majorised for short times by a corresponding solution of a Hamilton-Jacobi equation when the initial data is bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. This approach gives a remarkable identity for “connected parts” and accurate estimates which include criteria for convergence of iterated Mayer expansions.
D. C. Brydges

The Mean Field Bound for the Order Parameter of Bernoulli Percolation

We consider a general, translation invariant bond percolation model on ℤd with bonds characterized by couplings (Jx| X∈ℤ d ) and an inverse temperature parameter tf, with nontrivial critical value v c. We prove several inequalities including: (Da differential inequality for the infinite cluster density, P(v); and (2) an inequality relating the backbone density, Q(v), to p(v) and the expected size of finite clusters, χ’(v). If the above quantities exhibit critical scaling with exponents “defined” by P(v) ~ | vv c |β, Q(v) ~ | vv c |βQ, and χ’(v) ~ |vv c|-γ’ as vv c these inequalities imply the mean field bounds: β ≤ 1 and 2β ≤ βQ β + γ’. Furthermore, a magnetic backbone exponent, δQ, is defined analogously to the standard magnetic backbone exponent, δ. Again assuming critical scaling, our inequalities also imply the mean field bounds δ ≥ 2δQ and δQ ≥ 1.
J. T. Chayes, L. Chayes

Recent Results for the Stepping Stone Model

Let G be a graph, and C = {0,1,...,κ-1} a set of colors. We want to discuss a continuous time random process ζt with state space CG = configurations of colors from C on the graph G. This system ζt, known as the stepping stone model, has very simple dynamics: in any state ζ and at any site x, the color at that site waits a mean 1/2 exponential holding time and then paints a randomly chosen neighboring vertex with its color. We will also be treating a companion process ξt on {subsets of G} called coalescing random walks. As the name implies, ξt consists of rate 1/2 continuous time random walks on G which coalesce when they collide. Let us write ξ t A to denote the evolution of those walks which start on a subset A G. The graphs of principal interest to us are:
  • GN = the complete graph on {0,1,...,N-1},
  • Z N d = the d-dimensional integers with period N,
  • Zd = the d-dimensional integers.
Tropical interpretations of the dimension d are good therapy for a Minnesota February: Baja California (d=1), Society Islands (d=2), Caribbean Marine World (d=3), and of course Mathematical Physics (d > 4). The cardinality κ of the color set C might be 2 or 32,768 or even ∞.
J. Theodore Cox, David Griffeath

Stochastic Growth Models

This paper is based on a talk given by the first author at the I.M.A. in February, 1986 but incorporates improvements discovered during six later repiti-tions. The second authour should not be held responsible for the style of presentation of the results but should be given credit for discovering the results independently in the Fall of 1985. The discussion below is equal to the talk with most of the details of the proofs filled in, but we have tried to preserve the informal style of the talk and concentrate on the “main ideas” rather than giving complete details of the proofs. If we forget about definitions then the results can be summed up in a few words “Everything Durrett and Griffeath (1983) proved for one-dimensional nearest neighbor additive groth models is true for the corresponding class of finite range models, i.e., those which can be constructed from a percolation structure.”
R. Durrett, R. H. Schonmann

Random Walks and Diffusions on Fractals

We investigate the asymptotic motion of a random walker, which at time n is at X(n), on certain “fractal lattices”. For the “Sierpinski lattice” in dimension d we show that as → ∞, the process Y(t) ≡ X([(d+3) t])/2 converges in distribution (so that, in particular, |X(n)| ~ nγ, where γ = (ln 2)/ln(d + 3)) to a diffusion on the Sierpinski gasket, a Cantor set of Lebesgue measure zero. The analysis is based on a simple “renormalization group” type argument, involving self-similarity and “decimation invariance”.
Sheldon Goldstein

The Behavior of Processes with Statistical Mechanical Properties

Ever since Spitzer’s famous paper in 1970, there has been interest in a class of Markov processes which have as time-reversible stationary measures certain special distributions from the theory of statistical mechanics. The state space for these processes is \( \Xi = {\{ - 1, + 1\}^{{{Z^d}}}} \), which is the space of conf igurations of + and — spins on the sites of the lattice Z d . Transitions occur when there is a “flip” at a site x ∈ Z d, or in other words, a change of sign in the spin at x. The probability that a flip occurs at x in a short time interval (t, t + h], given the history of the process up to time t, is cxt)h + o(h), where ξt is the state of the process at time t, and cx is a nonnegative function defined on =, called the flip rate at x. Simultaneous flips at two different sites do not occur. A system of Markov processes with this description, me process for each possible initial state, is often called a “spin-flip system” with rates {cx}. Spitzer pointed out that for certain kinds of interaction potentials commonly used in statistical mechanics, one can always find a set of rates {cx} such that the corresponding spin-flip system has as time-reversible equilibria the Gibbs states that correspond to the interaction potential. (Spitzer’s results required a certain uniqueness hypothesis that was later verified for a large class of systems by Liggett (1972).)
Lawrence Gray

Stiff Chains and Levy Flight: Two Self Avoiding Walk Models and the Uses of Their Statistical Mechanical Representations

Two self-avoiding walk models are described: In the stiff chain model, useful in the description of some polymers, the walk “prefers” to continue in the direction of the last step with probability 1-p. The limit of small p and a large number of steps N is of particular interest. We present numerical results indicating the nature of the crossover from “ballistic” to random walk behavior. In two dimensions, the walk is asymptotically behaving like a self avoiding walk but in three dimensions, the numerical results suggest a random (not self-avoiding) behavior for a wide range of N. We describe this walk in terms of a vector spin model in the limit that the number of components n -> 0 and use this formulation to account for the difference between two and three dimensions observed numerically. Secondly, we consider self-avoiding Levy flight: the step length distribution is of the form P(x) = C/x μ+1. We study a type of Levy flight with a self-avoiding constraint called node-avoiding Levy flight here. This node-avoiding case is shown to be obtained from the n -> 0 limit of the statistical mechanics of another kind of vector spin model. The critical properties of this model were previously studied by Fisher, Nickel and Saks using the ε expansion. By comparing numerical simulations of node-avoiding Levy flight with their results, we can obtain information about the ε expansion in statistical mechanics at very small values of ε.
J. W. Halley

One Dimensional Stochastic Ising Models

We prove that for one dimensional stochastic Ising models with finite range interactions the rate of convergence to equilibrium is at least as fast as e-const t/log(t). There is no assumption here that the interaction must be attractive.
Richard Holley

A Scaling relation at criticality for 2D-Percolation

We prove a relation between the radius and volume of a two-dimensional percolation cluster. This implies for 2D percolation that the critical exponents δ and η satisfy η = 4/(δ + 1) (provided η exists).
Harry Kesten

Reversible Growth Models on Zd: Some Examples

In a recent paper [3], a class of reversible growth models on a fairly general set of sites was introduced and studied. These models are generalizations of the finite reversible nearest particle systems on the integers, which have been considered in several papers in recent years (see [1] and [2], for example). The focus of attention in these growth models is the probability of survival of the system. Typically there are natural one parameter families of models, and one wishes to determine the critical value for that parameter, which is the point at which survival with positive probability begins to occur. Once this is done, it is of interest to determine the manner in which the survival probability approaches its limit (which is usually zero) as the parameter approaches the critical value from above.
Thomas M. Liggett

Inequalities for γ and Related Critical Exponents in Short and Long Range Percolation

We relate the cluster size distribution Pn(p) at the percolation critical point, p = p|1c|0, to the critical exponent γ (ΣnPn(p) ≈ |pc — p| as p ↑ pc). If P(pc) > 0 (i.e., if P(p) is discontinuous at p = pc), then γ ≥ 2. If Pn(pc) ≈ n-1–1/δ as n → ∞, then γ ≥ 2(1 – 1/δ). Related inequalities are yalid for γr (ΣnrPn(p) ≈ |pc — p| r P as p ↑ pc) and γ r ' (defined analogously as p ↓ pc) when r > 1/δ: γr, γ r ' ≥ 2(r — 1/δ). These results are yalid for Bernoulli site or bond percolation on d-dimensional lattices for any d with p the site or bond occupation probability. They are also valid for long range translation invariant Bernoulli bond percolation with p the occupation probability for bonds of some given length.
C. M. Newman

A New Look at Contact Processes in Several Dimensions

We prove that if the complete convergence theorem holds for the basic contact process in dimension d with infection parameter λ larger than the critical value in this dimension, then the same theorem holds for this process in any dimension d’ > d for any λ’ > λ.
Roberto H. Schonmann

Fractal and Multifractal Approaches to Percolation: Some Exact and Not-So-Exact Results

I feel flattered to be included in such an impressive group of talks by mathematically gifted people. The last rigorous proof that was completely “my own” was in 1968 (Stanley 1968a) and this is now 1986. Hence in accepting Professor Kesten’s kind invitation I assume that my role is to unveil a set of percolation results that—unlike those in “the book” (Kesten 1981)— has not yet reached the stage of completely rigorous proofs. Accordingly, I shall focus mainly on relatively recent developments in percolation theory, some of which almost certainly may be amenable to exact analysis.
H. Eugene Stanley

Surface Simulations for Large Eden Clusters

Since the work with J.G. Zabolitzky on the Cray-2 at the University of Minnesota, from which I presented preliminary results at this conference, has been published in the meantime /1/, I now only given an overview, why simulations with millions of sites were needed to see the asymptotic behavior of the surface thickness of clusters in the Eden process. Also I add some remarks about still controversial points.
D. Stauffer

Duality for k-Degree Percolation on the Square Lattice

A generalization of the standard percolation problem, called k-degree percolation, is considered on the square lattice. In k-degree percolation, one investigates the probability of existence of an infinite path in which each vertex has k or more open edges incident to it.
One motivation for the study of k-degree percolation is to provide models which exhibit multiple phase transitions, since most substances exist in at least three possible phases. Quintas (1983) considered k-degree percolation on the ice lattice to estimate the volume function for water.
The duality approach to percolation problems on matching pairs of graphs is extended, establishing that the three standard critical probabilities -- P H ,P T , and p S -- are equal for k-degree percolation on the square lattice. Furthermorej the dual model for the 4-degree (or full-degree) model on the square lattice is shown to be equivalent to a standard percolation model on a nonplanar graph which is not a member of a matching pair of graphs in the sense of Sykes and Essam (1964), establishing that P H = P T = P S for this model. The approach may lead to the formulation oi a broader class of graphs for which this equality holds.
John C. Wierman
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