A shape theorem for the spread of an infection

• 成果类型: Article
• 署名作者: Kesten, Harry; Sidoravicius, Vladas
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.167.701
• 发表日期: 2008
• 页码: 701-766
• 关键词: growth

摘要

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 mu(A) Poisson distribution and that the N-A (x, 0-), x is an element of 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. [KSbJ gave some basic estimates for the growth of the set (B) over tilde (t) : = {x is an element of 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) over tilde (t) + [-1/2, 1/2](d) grows 2 2 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.