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

TL;DR

This paper analyzes the symmetric exclusion process in higher dimensions, showing that particle counts beyond a threshold become Poisson distributed over time, with explicit results for polynomial initial regions and extremal particle behavior.

Contribution

It establishes the convergence to Poisson distribution for particles beyond a threshold and explicitly characterizes extremal particle limits in symmetric exclusion.

Findings

Particle counts converge to Poisson distribution under certain conditions.

Explicit limits for polynomial growth initial regions.

Gumbel distribution for extremal particle positions.

Abstract

We consider the symmetric simple exclusion system on $\mathbb{Z}^d$, $d \ge 2$, 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 $\mathbb{Z}^d$ at time $t$. We show in large generality that when $\lim_{t\to\infty} E[N_t]$ exists, correlations between particles beyond $z$ vanish as $t \to \infty$ 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.