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

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.