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

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.