For symmetric exclusion on Z^d, d>=2, with step and polynomial-shape initial conditions, particle counts beyond a suitable superdiffusive level are asymptotically Poisson, implying Gumbel limits for the rightmost particle and all order statistics.
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Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1
For symmetric exclusion on Z^d, d>=2, with step and polynomial-shape initial conditions, particle counts beyond a suitable superdiffusive level are asymptotically Poisson, implying Gumbel limits for the rightmost particle and all order statistics.