Exponential rate for the contact process extinction time

TL;DR

This paper proves that for various random graph models, the extinction time of the contact process grows exponentially with the graph size when the infection rate exceeds the critical threshold.

Contribution

It establishes a universal exponential growth rate for the contact process extinction time across diverse random graph models in the supercritical regime.

Findings

Logarithm of extinction time divided by graph size converges to a positive constant.

Results apply to percolation models, random interlacements, Gaussian free field, and Galton-Watson trees.

Extinction time exhibits exponential growth in the size of the graph.

Abstract

We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.