Chase-escape on the configuration model

TL;DR

This paper investigates the chase-escape process on the configuration model, establishing bounds on the critical rate for red particles to occupy a positive fraction of vertices and identifying a phase transition in red occupation.

Contribution

It provides conjecturally sharp bounds on the critical rate _c and characterizes the phase transition in red occupancy for the chase-escape process on random graphs.

Findings

Derived lower and upper bounds on  for supercritical graphs

Identified the phase transition point for red occupation

Analyzed the behavior of red particles in the chase-escape process

Abstract

Chase-escape is a competitive growth process in which red particles spread to adjacent empty sites according to a rate-$\lambda$ Poisson process while being chased and consumed by blue particles according to a rate-$1$ Poisson process. Given a growing sequence of finite graphs, the critical rate $\lambda_c$ is the largest value of $\lambda$ for which red fails to reach a positive fraction of the vertices with high probability. We provide a conjecturally sharp lower bound and an implicit upper bound on $\lambda_c$ for supercritical random graphs sampled from the configuration model with independent and identically distributed degrees with finite second moment. We additionally show that the expected number of sites occupied by red undergoes a phase transition and identify the location of this transition.