Particle density in diffusion-limited annihilating systems

Abstract

Place an A-particle at each site of a graph independently with probability p, and otherwise place a B-particle. A- and B-particles perform independent continuous time random walks at rates ${\mathit{\lambda }}_{\mathit{A}}$ and ${\mathit{\lambda }}_{\mathit{B}}$, respectively, and annihilate upon colliding with a particle of opposite type. Bramson and Lebowitz studied the setting ${\mathit{\lambda }}_{\mathit{A}}={\mathit{\lambda }}_{\mathit{B}}$ in the early 1990s. Despite recent progress, many basic questions remain unanswered when ${\mathit{\lambda }}_{\mathit{A}}\ne {\mathit{\lambda }}_{\mathit{B}}$. For the critical case $\mathit{p}=1/2$ on low-dimensional integer lattices, we give a lower bound on the expected number of particles at the origin that matches physicists’ predictions. For the process with ${\mathit{\lambda }}_{\mathit{B}}=0$ on the integers and on the bidirected regular tree, we give sharp upper and lower bounds for the expected total occupation time of the root at and approaching criticality.