Abstract

In [KSb] we studied the following model for the spread of a rumor or infection: There is a “gas” of so-called A-particles, each of which performs a continuous time simple random walk on Z^d, with jump rate D_A. We assume that “just before the start” the number of A-particles at x, N_A(x,0−), has a mean μ_A Poisson distribution and that the N_A(x,0−), x in Z^d, are independent. In addition, there are B-particles which perform continuous time simple random walks with jump rate D_B. We start with a finite number of B-particles in the system at time 0. The positions of these initial B-particles are arbitrary, but they are nonrandom. The B-particles move independently of each other. The only interaction occurs when a B-particle and an A-particle coincide; the latter instantaneously turns into a B-particle. [KSb] gave some basic estimates for the growth of the set B(t) := {x in Z^d : a B-particle visits x during [0,t]}. In this article we show that if D_A = D_B, then B(t) := B(t) + [−,]^d grows linearly in time with an asymptotic shape, i.e., there exists a nonrandom set B₀ such that (1 ∕ t)B(t) → B₀, in a sense which will be made precise.

Mathematical Subject Classification

Primary: 60K35

Authors