# The contact process on the regular tree with random vertex weights

## TL;DR

This paper investigates the contact process with random vertex weights on large regular trees, identifying phase transitions in infection survival and their asymptotic behavior as the degree increases.

## Contribution

It introduces a model with random vertex weights on regular trees and analyzes the asymptotic behavior of critical infection rates as the degree grows.

## Key findings

- Existence of a phase transition at a critical infection rate $\\lambda_c(d)$.
- Identification of another phase transition at $\\lambda_e(d)$ for exponential die-out.
- Asymptotic equivalence of the two critical values as degree increases.

## Abstract

This paper is concerned with contact process with random vertex weights on regular trees, and study the asymptotic behavior of the critical infection rate as the degree of the trees increasing to infinity. In this model, the infection propagates through the edge connecting vertices $x$ and $y$ at rate $\lambda\rho(x)\rho(y)$ for some $\lambda>0$, where $\{\rho(x),{x\in T^d}\}$ are $i.i.d.$ vertex weights. We show that when $d$ is large enough there is a phase transition at $\lambda_c(d)\in(0,\infty)$ such that for $\lambda<\lambda_c(d)$ the contact process dies out, and for $\lambda>\lambda_c(d)$ the contact process survives with a positive probability. Moreover, we also show that there is another phase transition at $\lambda_e(d)$ such that for $\lambda<\lambda_e(d)$ the contact process dies out at an exponential rate. Finally, we show that these two critical values have the same asymptotic behavior as $d$ increases.