Cover levels and random interlacements

TL;DR

This paper studies the cover levels of finite sets in the random interlacements model, showing that as the set size grows, the rescaled cover level converges in distribution to a Gumbel distribution, revealing a universal extreme value behavior.

Contribution

It establishes the asymptotic distribution of cover levels in the random interlacements model, connecting it to extreme value theory and the Gumbel distribution.

Findings

Rescaled cover levels converge to Gumbel distribution as set size increases.

Provides a probabilistic characterization of cover times in the model.

Extends understanding of extremal properties in random interlacements.

Abstract

This note investigates cover levels of finite sets in the random interlacements model introduced in [Ann. of Math. (2) 171 (2010) 2039-2087], 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 $\operatorname {exp}(-\operatorname {exp}(-z))$.