Universality of cutoff for the Exclusion Process

Abstract: Under a fairly general condition on the underlying jump rates, we prove that the Exclusion Process with fixed particle density exhibits cutoff at time $t_{mix}\sim \frac{1}{2}{t}<em>{rel} \log N$, where $N$ is the volume and ${t}</em>{rel}$ the relaxation time of the single-particle dynamics. Our result covers, in particular, the standard setting of uniform nearest-neighbor jumps on large discrete tori in any fixed dimension and, more generally, on any sequence of vertex-transitive graphs with bounded degree and polynomially diverging diameter. Our (short and entirely human) proof combines Wilson's method, the Octopus Inequality, Fourier analysis on Hamming slices, and a sharp comparison with the Dirichlet form of a natural accelerated variant of the dynamics.