Critical first passage percolation on random graphs

TL;DR

This paper investigates the behavior of first passage percolation on large random graphs generated by a supercritical configuration model, focusing on the conditions under which passage times between vertices remain bounded or diverge as the network grows.

Contribution

It extends Zhang's critical percolation results to random graph models, providing new insights into passage time limits in the supercritical regime.

Findings

Passage times can remain bounded or diverge depending on edge weight distribution.

The study identifies conditions for the convergence in distribution of passage times.

Results highlight differences between lattice and random graph percolation behaviors.

Abstract

In 1999, Zhang proved that, for first passage percolation on the square lattice $\mathbb{Z}^2$ with i.i.d. non-negative edge weights, if the probability that the passage time distribution of an edge $P(t_e = 0) =1/2 $, the critical value for bond percolation on $\mathbb{Z}^2$, then the passage time from the origin $0$ to the boundary of $[-n,n]^2$ may converge to $\infty$ or stay bounded depending on the nature of the distribution of $t_e$ close to zero. In 2017, Damron, Lam, and Wang gave an easily checkable necessary and sufficient condition for the passage time to remain bounded. Concurrently, there has been tremendous growth in the study of weak and strong disorder on random graph models. Standard first passage percolation with strictly positive edge weights provides insight in the weak disorder regime. Critical percolation on such graphs provides information on the strong disorder (namely the minimal spanning tree) regime. Here we consider the analogous problem of Zhang but now for a sequence of random graphs $\{G_n:n\geq 1\}$ generated by a supercritical configuration model with a fixed degree distribution. Let $p_c$ denote the associated critical percolation parameter, and suppose each edge $e\in E(G_n)$ has weight $t_e \sim p_c \delta_0 +(1-p_c)\delta_{F_\zeta}$ where $F_\zeta$ is the cdf of a random variable $\zeta$ supported on $(0,\infty)$. The main question of interest is: when does the passage time between two randomly chosen vertices have a limit in distribution in the large network $n\to \infty$ limit? There are interesting similarities between the answers on $\mathbb{Z}^2$ and on random graphs, but it is easier for the passage times on random graphs to stay bounded.