Oriented first passage percolation in the mean field limit, 2. The extremal process

TL;DR

This paper proves that the extremal process of oriented first passage percolation on high-dimensional hypercubes converges to a Cox process with exponential intensity, revealing universal distributional limits and correlation effects.

Contribution

It establishes the convergence of the extremal process to a Cox process with explicit distribution, advancing understanding of high-dimensional percolation behavior.

Findings

Extremal process converges to Cox process with exponential intensity.

First passage time converges to a shifted Gumbel distribution.

The shift distribution is universal and related to Bessel functions.

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.