Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1

2025/01/17 by Michael J. Conroy, Conroy, Michael, Sunder Sethuraman +1 · 1 citation
Mathematics · #60F05 #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2501.10522

openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the symmetric simple exclusion system on ℤ^d, d ≥ 2, starting from a class of ``step'' initial conditions in which particles are constrained within a half-space. One may count the number N_t of particles that have moved beyond a distance z = z(t) into the initially-empty half of ℤ^d at time t. We show in large generality that when lim_t→∞ E[N_t] exists, correlations between particles beyond z vanish as t → ∞ so as to allow convergence of N_t to the same Poisson distribution one would get were the particles allowed to move independently. When the initial condition constrains a region of polynomial growth, we identify z(t) and the limit of E[N_t] explicitly. As a consequence of the limit, we obtain a Gumbel limit distribution for the extremal particle position, as well as the limiting distributions of all order statistics.

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