Cover levels and random interlacements
The random interlacements model was introduced by \textit{A.-S. Sznitman} [Ann. Math. (2) 171, No. 3, 2039--2087 (2010; Zbl 1202.60160)]. It is a certain Poisson point process indexed by time \(u\) and taking values in the set of doubly infinite random walk paths (modulo time shift) on the \(d\)-dimensional integer lattice (\(d \geq 3\)). Denote by \(I^u\) the union of paths of this process up to time \(u\). For any finite subset \(A\) of the integer lattice, denote by \(M(A)\) the cover time of this process, that is, the smallest \(u\) such that \(A\subset I^u\). Finally, let \(g\) denote the Green function of a random walk on the \(d\)-dimensional integer lattice.NEWLINENEWLINEThe author shows that \(M(A)/g(0)\log|A|\) converges to \(1\) as the cardinality of \(A\) increases to infinity. More precisely, he shows that, for all finite \(A\), NEWLINE\[NEWLINE\sup_{z\in\mathbb{R}}\left|\operatorname{P}\left[\frac{M(A)}{g(0)}-\log(|A|)\,\leq z\right]-\exp\big(-e^{-z}\big)\right|\leq c|A|^{-c_1}NEWLINE\]NEWLINE for some \(c_1>0\) which is given more or less explicitly. Presumably, \(c\) denotes a constant that is independent of \(A\). In particular, \(M(A)/g(0)-\log(|A|)\) converges to the Gumbel distribution as \(|A|\to\infty\).
scientific article; zbMATH DE number 6034163
scientific article