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April 2020 Oriented first passage percolation in the mean field limit, 2. The extremal process
Nicola Kistler, Adrien Schertzer, Marius A. Schmidt
Ann. Appl. Probab. 30(2): 788-811 (April 2020). DOI: 10.1214/19-AAP1515

Abstract

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.

Citation

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Nicola Kistler. Adrien Schertzer. Marius A. Schmidt. "Oriented first passage percolation in the mean field limit, 2. The extremal process." Ann. Appl. Probab. 30 (2) 788 - 811, April 2020. https://doi.org/10.1214/19-AAP1515

Information

Received: 1 August 2018; Revised: 1 June 2019; Published: April 2020
First available in Project Euclid: 8 June 2020

zbMATH: 07236134
MathSciNet: MR4108122
Digital Object Identifier: 10.1214/19-AAP1515

Subjects:
Primary: 60G70, 60J80, 82B44

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 2 • April 2020
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