Limiting Speed and Fluctuations for the Boundary Modified Contact Process

TL;DR

This paper analyzes the boundary modified contact process, establishing laws of large numbers, central limit theorems, and tail bounds for the spread of infection, extending previous results and solving an open problem.

Contribution

It provides new probabilistic results for the boundary modified contact process, including laws of large numbers, CLTs, and tail bounds, for the first time.

Findings

Strong law of large numbers for the rightmost infected vertex

Central limit theorem for the location of the rightmost infected vertex

Stretched exponential tail bounds for fluctuations and extinction time

Abstract

The boundary modified contact process models an epidemic spreading in one dimension with two infection parameters, $\lambda_i$ and $\lambda_e$. Starting from a finite infected set, each edge of $\mathbb{Z}$ transmits the infection at rate $\lambda_i$ except for the rightmost and leftmost edges incident to infected vertices, which transmit the infection at rate $\lambda_e$. We show a strong law of large numbers and central limit theorem for the location of the rightmost infected vertex when $\lambda_i = \lambda_c$ and $\lambda_e = \lambda_c + \varepsilon$. We also show stretched exponential tail bounds in the fluctuations of the rightmost infected vertex, the extinction time of the process on the event of non-survival, and the probability of survival given the size of the initial infected region. Our results extend to the boundary modified contact process whenever $\lambda_c \leq \lambda_i < \lambda_e$, and solves an open problem first proposed by Andjel and Rolla in [1].