On the Growth of One Dimensional Contact Processes

Abstract

In this paper we will study the number of particles alive at time $t$ in a one dimensional contact process $\xi^0_t$ which starts with one particle at 0 at time 0. In the case of a nearest neighbor interaction we will show that if $|\xi^0_t|$ is the number of particles and $r_t, l_t$ are the positions of the rightmost and leftmost particles (with $r_t = l_t = 0$ if $|\xi^0_t| = 0$) then there are constants $\gamma, \alpha$, and $\beta$ so that $|\xi^0_t|/t, r_t/t$, and $l_t/t$ converge in $L^1$ to $\gamma 1_\Lambda, \alpha 1_\Lambda$ and $\beta 1_\Lambda$ where $\Lambda = \{|\xi^0_t| > 0$ for all $t\}$. The constant $\gamma = \rho(\alpha - \beta)^+$ where $\rho$ is the density of the "upper invariant measure" $\xi^Z_\infty$.