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Gumbel fluctuations for cover times in the discrete torus. (English) Zbl 1295.60053
For \(N\geq 3\) and \(d\geq 3\), the author considers a continuous time simple random walk\( (Y_{t})_{t\geq 0}\) in the torus \(T_{N}=(\mathbb{Z}/N\mathbb{Z})^{d}\) starting from the uniform distribution. Let \(H_{x}=\inf{t\geq 0:Y_{t}=x}\) denote the entrance time of a vertex \(x\in T_{N}\), and let \(C_{F}=\max_{x\in F}H_{x}\) define the cover time of a set \(F\subset T_{N}\). By constructing a coupling of \((Y_{t})_{t\subset 0}\) and independent random interlacements, the author proves that, for all \(F\subset T_{N}\) and some constant \(c>0, \sup_{z\in \mathbb{R}}|P(C_{F}\leq g(0,0)N^{d}(\text{log}|F|+z))-e^{-e^{-z}}|\leq c|F|^{-c}\), where g stands for the \(\mathbb{Z}^{d}\) Green function. This implies that \(C_{T_{N}}/g(0,0)N^{d}-\text{log}N^{d}\) converges in law to the standard Gumbel distribution as \(N\rightarrow \infty\).
Reviewer’s remark: Unfortunately, this paper is negligently and disorderly written. It also suffers from excessive verbosity and extended awkwardness. Moreover, errors such as “a point processes”, “variance of Var”, “the event the event”, “and and” are encountered.

MSC:
60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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