A shape theorem for the spread of an infection

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 ∈ 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 ∈ 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) + [-1 2 , 1 2 ] d grows linearly in time with an asymptotic shape, i.e., there exists a nonrandom set B 0 such that (1/t)B(t) → B 0 , in a sense which will be made precise.

Locations

• arXiv (Cornell University) - View - PDF
• Annals of Mathematics - View - PDF