On the universality of fluctuations for the cover time

Abstract

We consider random walks on vertex-transitive graphs of bounded degree. We show that subject to a simple diameter condition (which guarantees in particular that the walk is in some sense locally transient), the cover time fluctuations are universal: after rescaling, they converge to a standard Gumbel distribution. We further show by constructing an explicit counter-example that our diameter condition is sharp in a very strong sense. Surprisingly, this counter-example is also locally transient. We complement our result by showing that near the cover time, the distribution of the uncovered set is close in total variation to a product measure. The arguments rely on recent breakthroughs by Tessera and Tointon on finitary versions of Gromov's theorem on groups of polynomial growth, which are leveraged into strong heat kernel bounds that imply decorrelation of the uncovered set. Another key aspect is an improvement on the exponential approximation of hitting times due to Aldous and Brown, which is of independent interest.

Cover-time Gumbel fluctuations in finite-range, symmetric, irreducible random walks on torus

“…Remark 1. Recently, Berestycki et al [29] proved under a broader setting the Gumbel fluctuations of random walks on general finite vertex-transitive graphs using a different approach relying on a finitary version of Gromov's theorem [30]. Our work is independent of theirs and is more of a natural generalization of [24] alongside the random interlacements approach.…”