# 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.

## Key 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.

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Source: https://tomesphere.com/paper/0908.4146