Phase transition for accessibility percolation on hypercubes

TL;DR

This paper investigates the phase transition in accessibility percolation on hypercubes, establishing a sharp threshold for the existence of increasing paths between vertices, thus resolving a conjecture from 2014.

Contribution

It provides a complete characterization of the phase transition and critical window for accessibility percolation on hypercubes, confirming a longstanding conjecture.

Findings

Identified the critical threshold for the phase transition.

Determined the precise critical window.

Resolved the conjecture of Berestycki, Brunet and Shi (2014).

Abstract

In this paper, we consider accessibility percolation on hypercubes, i.e., we place i.i.d. uniform [0,1] random variables on vertices of a hypercube, and study whether there is a path connecting two vertices such that the values of these random variables increase along the path. We establish a sharp phase transition depending on the difference of the values at the two endpoints, and determine the critical window of the phase transition. Our result completely resolves a conjecture of Berestycki, Brunet and Shi (2014).