Cutoff for random walk on dynamical Erdős–Rényi graph

We consider dynamical percolation on the complete graph Kn, where each edge refreshes its state at rate μ 1/n, and is then declared open with probability p = λ/n where λ > 1. We study a random walk on this dynamical environment which jumps at rate 1/n along every open edge. We show that the mixing time of the full system exhibits cutoff at logn/μ. We do this by showing that the random walk component mixes faster than the environment process; along the way, we control the time it takes for the walk to become isolated.

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