A probabilistic proof of cutoff in the Metropolis algorithm for the Erd\H{o}s-R\'enyi random graph

We study mixing of the Metropolis algorithm for a distribution on the hypercube that corresponds to the Erd\H{o}s-R\'enyi random graph with edge probability p. This Markov chain has cutoff at max{p,1-p} n log n with window size n, a result proved by Diaconis and Ram (2000) using Fourier analysis. Here we give an alternative proof that relies on coupling and a projection to a two-dimensional Markov chain. This is done in the hope that probabilistic techniques will be easier to generalize to less symmetric distributions. We also describe a close relationship between the Metropolis and Gibbs samplers for this model. Our proof extends to the case where the edge probabilities vary with n. In that case, we also show that a natural coordinate wise coupling is sharp if and only if the edge probabilities are of order 1/n.