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Oriented first passage percolation in the mean field limit. II: The extremal process. (English) Zbl 1464.60053

Summary: This is the second, and last paper in which we address the behavior of oriented first passage percolation on the hypercube in the limit of large dimensions. We prove here that the extremal process converges to a Cox process with exponential intensity. This entails, in particular, that the first passage time converges weakly to a random shift of the Gumbel distribution. The random shift, which has an explicit, universal distribution related to modified Bessel functions of the second kind, is the sole manifestation of correlations ensuing from the geometry of Euclidean space in infinite dimensions. The proof combines the multiscale refinement of the second moment method with a conditional version of the Chen-Stein bounds, and a contraction principle.
For Part I, see [the authors, Braz. J. Probab. Stat. 34, No. 2, 414–425 (2020; Zbl 1453.60159)].

MSC:

60G70 Extreme value theory; extremal stochastic processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C80 Random graphs (graph-theoretic aspects)
82B43 Percolation

Citations:

Zbl 1453.60159
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References:

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