On the universality of fluctuations for the cover time · arXivDesk
2202.02255Feb 4, 2022v4:57 pages. Version revised following reviewer's comments. In particular, the improvement of the Aldous-Brown estimates on the exponential approximation for hitting times of sets have been separated from this article and can instead be found in arXiv:2601.03864. Section 3 in particular has also been restructured
On the universality of fluctuations for the cover time
Nathanaël Berestycki, Jonathan Hermon, Lucas Teyssier
We consider random walks on finite vertex-transitive graphs Γ of bounded degree. We find a simple geometric condition which characterises the cover time fluctuations: the suitably normalised cover time converges to a standard Gumbel variable if and only if Diam(Γ)2=o(n/logn), where n=∣Γ∣
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. We prove that this condition is furthermore equivalent to the decorrelation of the uncovered set. The arguments rely on recent breakthroughs by Tessera and Tointon on finitary versions of Gromov's theorem on groups of polynomial growth, which we leverage into strong heat kernel bounds, and refined quantitative estimates on Aldous and Brown's exponential approximation of hitting times, which are of independent interest.