A shape theorem for the spread of an infection. (English) Zbl 1202.92077

Kesten, Harry; Sidoravicius, Vladas

Ann. Math. (2) 167, No. 3, 701-766 (2008).

This paper sharpens our former results in Ann. Probab. 33, No. 6, 2402–2462 (2005; Zbl 1111.60074), for the following model for the spread of a rumor or infection: There is a “gas” of so-called A-particles, each which performs a continuous time simple random walk on \(\mathbb{Z}^d\), with jump rate \(D_A\). The number of A-particles at \(x\) just before the start, \(N_A(x,0-)\), are mutually independent over \(x\) and have a mean \(\mu_A\) Poisson distribution. In addition there are B-particles which perform continuous time simple random walks with jump rate \(D_B\). A finite number of B-particles are started in the system at time 0. The positions of these initial B-particles are arbitrary and non-random. The B-particles move independent of each other. The only interaction occurs when a B-particle and an A-particle coincide; the latter instantaneously turns into a B-particle.
The paper studies the growth of the set \(\bar{B}(t) = \{ x \in \mathbb{Z}^d : \text{a B-particle visits }x\) during \([0,t]\)

Reviewer: Bo Markussen (Kopenhagen)

MSC:

92D30	Epidemiology
60K35	Interacting random processes; statistical mechanics type models; percolation theory
82C22	Interacting particle systems in time-dependent statistical mechanics
60G50	Sums of independent random variables; random walks

Keywords:

interacting particles; random walk; halfspaces; superconvolutivity; frog model

Citations:

Zbl 1111.60074; Zbl 1013.60081