On the universality of fluctuations for the cover time
| Autor: | Berestycki, Nathanaël, Hermon, Jonathan, Teyssier, Lucas |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider random walks on finite vertex-transitive graphs $\Gamma$ 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 $\mathrm{Diam}(\Gamma)^2 = o(n/\log n)$, where $n = |\Gamma|$. 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. Comment: v3: 70 pages. The main theorem has been considerably strengthened: we now show that the diameter condition is necessary *and* sufficient for convergence of the rescaled cover time to a Gumbel distribution, and is also equivalent to the decorrelation (in the total variation sense) of the uncovered set. The presentation has also been thoroughly revised and streamlined. v2: 58 pages |
| Databáze: | arXiv |
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