- Gumbel fluctuations for cover times in the discrete torus (Submitted.)
[ Abstract, pdf ]This work proves that the fluctuations of the cover time of simple random walk in
the discrete torus of dimension at least three with large side-length are governed
by the Gumbel extreme value distribution. This result was conjectured for example in
the book by Aldous & Fill. We also derive some corollaries which qualitatively describe
"how" covering happens. To enable the proof we develop a new and stronger coupling
of the model of random interlacements, introduced by Sznitman, and random walk in
the torus. This coupling is also of independent interest.
- Cover times in the discrete cylinder (Submitted.)
[ Abstract, pdf ] This article proves that, in terms of local times, the properly rescaled and re-centered
cover times of finite subsets of the discrete cylinder by simple random walk
converge in law to the Gumbel distribution, as the cardinality of the set goes to
infinity. As applications we obtain several other results related to covering in the
discrete cylinder. Our method is new and involves random interlacements, which
were introduced in [22]. To enable the proof we develop a new stronger coupling
of simple random walk in the cylinder and random interlacements, which is also of
independent interest.
- Cover levels and random interlacements (accepted for publication in the Annals of Applied Probability.)
[ Abstract, pdf ]This note investigates cover levels of finite sets in the random interlacements model, that is the least level such that the set is completely contained in the random interlacement at that level. It proves that as the cardinality of a set goes to infinity, the rescaled and recentered cover level tends in distribution to the Gumbel distribution with cumulative distribution function exp(-exp(-z)).
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