First passage percolation in the mean field limit

TL;DR

This paper provides a streamlined proof for Aldous' conjecture on first passage percolation in the hypercube's mean field limit, utilizing ideas from Derrida's random energy models.

Contribution

It offers a more natural and simplified proof of Aldous' conjecture, improving upon previous variance reduction methods.

Findings

Confirmed Aldous' conjecture in the mean field limit

Developed a new proof based on Derrida's random energy models

Simplified the understanding of first passage percolation in high dimensions

Abstract

The Poisson clumping heuristic has lead Aldous to conjecture the value of the first passage percolation on the hypercube in the limit of large dimensions. Aldous' conjecture has been rigorously confirmed by Fill and Pemantle [Annals of Applied Prob- ability 3 (1993)] by means of a variance reduction trick. We present here a streamlined and, we believe, more natural proof based on ideas emerged in the study of Derrida's random energy models.