The extinction of the contact process in a one-dimensional random environment with long-range interactions

TL;DR

This paper investigates the extinction behavior of the contact process on a one-dimensional long-range percolation cluster, establishing conditions under which the process dies out based on transmission rates.

Contribution

It introduces a renormalization and Peierls-type argument approach to analyze contact process extinction in a long-range percolation environment.

Findings

Contact process dies out below a critical transmission rate

Application of renormalization and Peierls-type methods

Control of crossing probabilities in randomly-stretched lattices

Abstract

We study the contact process on the long-range percolation cluster on $\mathbb{Z}$ where each edge $\langle i,j \rangle$ is open with probability $|i-j|^{-s}$ for $s> 2$. Using a renormalization procedure we apply Peierls-type argument to prove that the contact process dies out if the transmission rate is smaller than a critical threshold. Our methods involve the control of crossing probabilities for percolation on randomly-stretched lattices as in https://doi.org/10.1214/22-AAP1887.