Asymptotic Behavior of Critical Infection Rates for Threshold-one Contact Processes on Lattices and Regular Trees

2013/11/29 by Xiaofeng Xue, Xue, Xiaofeng
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1311.7448

arxiv created 2013/11/29 · arxiv updated 2013/12/02

Abstract

In this paper we study threshold-one contact processes on lattices and regular trees. The asymptotic behavior of the critical infection rates as the degrees of the graphs growing to infinity are obtained. Defining λ_c as the supremum of infection rates which causes extinction of the process at equilibrium, we prove that nλ_c^Tn→1 and 2dλ_c^Zd→1 as n,d→+∞. Our result is a development of the conclusion that λ_c^Zd≤(2.18)/(d) shown in \citeDur1991. To prove our main result, a crucial lemma about the probability of a simple random walk on a lattice returning to zero is obtained. In details, the lemma is that lim_d→+∞2dP(∃ n≥1, S_n^(d)=0)=1, where S_n^(d) is a simple random walk on Z^d with S₀^(d)=0.

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