J. Theodore Cox

Large deviations for Poisson systems of independent random walks

SummaryWe prove large deviation theorems for occupation time functionals of independent random walks started from a Poisson field on Zd. In dimensions 1 and 2 the large deviation tails are larger than exponential. Exact asymptotics are derived.

Oriented percolation in dimensions d ≥ 4: bounds and asymptotic formulas

Let p c (d) be the critical probability for oriented percolation in ℤ d and let μ(d) be the time constant for the first passage process based on the exponential distribution. In this paper we show that as d → ∞, dp c (d) and d μ,( d ) → γ where γ is a constant in [e −1 , 2 −1 ] which we conjecture to be e −1 . In the case of p c (d) we have made some progress toward obtaining an asymptotic expansion in powers of d −1 . Our results show The left hand side agrees, up to O(d −3 ) , with a (nonrigorous) series expansion of Blease (1, 2):

Comparison of interacting diffusions and an application to their ergodic theory

SummaryA general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.

THE STEPPING STONE MODEL. II: GENEALOGIES AND THE INFINITE SITES MODEL

This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the two-dimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman’s coalescent. With DNA sequence data in mind, we investigate mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site. Our results suggest that the spatial structure of the human population contributes to the haplotype structure and a slower than expected decay of genetic correlation with distance revealed by recent studies of the human genome.

Rescaled Lotka-Volterra Models Converge to Super-Brownian Motion

We show that a sequence of stochastic spatial Lotka-Volterra models, suitably rescaled in space and time, converges weakly to super-Brownian motion with drift. The result includes both long range and nearest neighbor models, the latter for dimensions three and above. These theorems are special cases of a general convergence theorem for perturbations of the voter model.

Recurrence and Ergodicity of Interacting Particle Systems

Preamble. An important technical result (Proposition 2.3) and its proof, present in the original submission, was erroneously omitted when this paper was published in 2000. The missing text, which should have appeared on page 243 directly before Section 3, is included in this erratum, together with a short account of its context. Simultaneously with the publication of this erratum, the electronic version of the paper in Vol. 116 No. 2 (2000) will be completed by insertion of the missing text. Readers of Probability Theory and Related Fields who have access to the electronic version of this erratum will also have access via a URL to the full, intact paper.

Occupation time large deviations of the voter model

This paper is a sequel to [5] and [6]. We continue our study of occupation time large deviation probabilities for some simple infinite particle systems by analysing the so-called voter model ζt (see e.g., [11] or [8]). In keeping with our previous results, we show that the large deviations are “classical” in high dimensions (d≧5 for ζt) but “fat” in low dimensions (d≦4). Interaction distinguishes the voter model from the independent particle systems of [5] and [6], and consequently exact computations no longer seem feasible. Instead, we derive upper and lower bounds which capture the asymptotic decay rate of the large deviation tails.

A duality relation for entrance and exit laws for Markov processes

Markov processes Xt on (X, FX) and Yt on (Y, FY) are said to be dual with respect to the function f(x, y) if Exf(Xt, y) = Eyf(x, Yt for all x [epsilon] X, y [epsilon] Y, t [greater-or-equal, slanted] 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1{x

Consolidation rates for two interacting systems in the plane

SummaryThis paper is a sequel of a paper of Cox and Griffeath “diffusive clustering in the two dimensional voter model”. We continue our study of the voter model and coalescing random walks on the two dimensional integer lattice. Some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived. We use these results to prove a limit law, announced in that earlier paper, concerning the size of the largest square centered at the origin which is of solid color at a large time t.

Survival and coexistence for a multitype contact process

We study the ergodic theory of a multitype contact process with equal death rates and unequal birth rates on the d-dimensional integer lattice and regular trees. We prove that for birth rates in a certain interval there is coexistence on the tree, which by a result of Neuhauser is not possible on the lattice. We also prove a complete convergence result when the larger birth rate falls outside of this interval.