Lawrence Gray

A reader's guide to Gacs's "positive rates" paper

Peter Gacss monograph, which follows this article, provides a counterexample to the important Positive Rates Conjecture. This conjecture, which arose in the late 1960s, was based on very plausible arguments, some of which come from statistical mechanics. During the long gestation period of the Gacs example, there has been a great deal of skepticism about the validity of his work. The construction and verification of Gacss counterexample are unavoidably complex, and as a consequence, his paper is quite lengthy. But because of the novelty of the techniques and the significance of the result, his work deserves to become widely known. This readers guide is intended both as a cheap substitute for reading the whole thing, as well as a warm-up for those who want to plumb its depths.

The survival of branching annihilating random walk

SummaryBranching annihilating random walk is an interacting particle system on ℤ. As time evolves, particles execute random walks and branch, and disappear when they meet other particles. It is shown here that starting from a finite number of particles, the system will survive with positive probability if the random walk rate is low enough relative to the branching rate, but will die out with probability one if the random walk rate is high. Since the branching annihilating random walk is non-attractive, standard techniques usually employed for interacting particle systems are not applicable. Instead, a modification of a contour argument by Gray and Griffeath is used.

A Useful Renormalization Argument

We define a collection of ‘generic’ population models for which we prove a survival criterion by using a renormalization argument. These models can be compared with other more familiar models, leading to simple proofs of various survival results. In particular, we prove a generalization of Toom’s Theorem concerning survival in multidimensional probabilistic cellular automata. Our technique should also be applicable to a variety of other discrete and continuous time models.

The positive rates problem for attractive nearest neighbor spin systems on ℤ

SummaryIt is shown that a spin system on ℤ has only one invariant probability measure if it has attractive or repulsive nearest neighbor flip rates which are strictly positive and periodic under translation along ℤ.

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 cx(ξt)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).)

Lower bounds for the critical probability in percolation models with oriented bonds

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.

Phase Transitions in Traffic Models

It is suggested that the question of existence of a jamming phase transition in a broad class of single-lane cellular-automaton traffic models may be studied using a correspondence to the asymmetric chipping model. In models where such correspondence is applicable, jamming phase transition does not take place. Rather, the system exhibits a smooth crossover between free-flow and jammed states, as the car density is increased.

On the uniqueness of certain interacting particle systems

We discuss the existence, uniqueness and ergodicity of certain configurationvalued stochastic processes (it)t~+. Let V be a countable set of vertices, or sites, and S a compact metric space. At each time t our process assumes a value i, ~ S = SV; it is to be thought of as a configuration of values from S on the sites of V, i~(x) being the value at x. Let ID=ID(1R +, S) be the path space of right continuous functions with left limits from [0, oo) to & define ~t: ID--* S as the evaluation map co ~ co(t), co =(cot(x))~lD, and let ~ = a <(~t)t~§ be the a-algebra generated by the it. Also, put N~)=a<(~r)o=<,=<t). We view the desired stochastic system as the coordinate process (~) on (ID, 2) governed by any of a collection (P~)~ of probability measures such that

Toom’s Stability Theorem in Continuous Time

This paper provides a continuous-time analogue of Toom’s famous discrete-time stability criterion. Because of certain intrinsic differences between discrete and continuous time, simple analogues of Toom’s result are not true, the main problem being that discrete-time models can undergo a spatial shift at each time step and continuous-time models cannot. In our main result, we show that once such shifts have been neutralized, the stability properties of a discrete-time model and its continuous-time analogue are the same. This result applies to a large class of models with finite range interactions in finite dimensions, including many for which the stability question was previously unanswered. Its proof uses an improved version of Toom’s theorem that is found in [BG91]. We also obtain, as a byproduct of our analysis, an alternative criterion for stability in discrete time that is easy to check.

Extreme paths in oriented two-dimensional percolation

A useful result about leftmost and rightmost paths in two dimensional bond percolation is proved. This result was introduced without proof in \cite{G} in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete time contact process and two dimensional site percolation. Among the consequences are a natural monotonicity in the probability of percolation between different sites and a somewhat counter-intuitive correlation inequality.