The survival of large dimensional threshold contact processes

TL;DR

This paper investigates the behavior of large-dimensional threshold contact processes and voter models, showing that the critical infection threshold diminishes as dimension increases, leading to nondegenerate invariant measures.

Contribution

It demonstrates that the critical infection parameter for survival vanishes as the dimension grows, revealing new insights into high-dimensional contact processes and voter models.

Findings

Critical point $\lambda_c$ tends to zero as $d o \infty$

Threshold voter model has a nondegenerate extremal invariant measure in high dimensions

Survival probability behavior changes with increasing dimension

Abstract

We study the threshold $\theta$ contact process on $\mathbb{Z}^d$ with infection parameter $\lambda$. We show that the critical point $\lambda_{\mathrm{c}}$, defined as the threshold for survival starting from every site occupied, vanishes as $d\to\infty$. This implies that the threshold $\theta$ voter model on $\mathbb{Z}^d$ has a nondegenerate extremal invariant measure, when $d$ is large.