Limiting Speed and Fluctuations for the Boundary Modified Contact Process

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 ${\lambda }_e={\lambda }_c+\epsilon$. 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 ${\lambda }_c\le {\lambda }_i<{\lambda }_e$, and solves an open problem first proposed by Andjel and Rolla in [1].