Metastability for the contact process on the configuration model with infinite mean degree

TL;DR

This paper investigates the metastability and extinction times of the contact process on the configuration model with infinite mean degree, showing exponential growth and convergence properties as the graph size increases.

Contribution

It extends previous metastability results to the case of power law degree distributions with exponent less than or equal to two.

Findings

Extinction time grows exponentially with graph size.

Extinction time divided by its mean converges to an exponential distribution.

Density of infected sites converges to a constant at exponential times.

Abstract

We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two. We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an exponent larger than $2$ obtained in \cite{CD,MMVY,MVY}.