A simple proof of exponential decay of subcritical contact processes

TL;DR

This paper presents a new, simplified proof demonstrating exponential decay in subcritical contact processes on lattices, providing explicit bounds and insights into the critical exponent and survival probabilities.

Contribution

It introduces a straightforward proof technique for exponential decay and derives explicit bounds on survival probability below the critical recovery rate.

Findings

Expected number of infected sites decays exponentially in subcritical regime

Explicit bounds on survival probability below critical recovery rate

Critical exponent is bounded from below by mean-field value

Abstract

This paper gives a new, simple proof of the known fact that for contact processes on general lattices, in the subcritical regime the expected number of infected sites decays exponentially fast as time tends to infinity. The proof also yields an explicit bound on the survival probability below the critical recovery rate, which shows that the critical exponent associated with this function is bounded from below by its mean-field value. The main idea of the proof is that if the expected number of infected sites decays slower than exponentially, then this implies the existence of a harmonic function that can be used to show that the process survives for any lower value of the recovery rate.