Threshold theta geq 2 contact processes on homogeneous trees

2006/03/04 by Luiz Renato Fontes, Roberto H. Schonmann, Fontes, Luiz Renato +1
Mathematics · #60K35 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60K35

paper · pdf · doi:10.48550/arxiv.math/0603109

27 pages

arxiv created 2006/03/04 · arxiv updated 2009/12/01

Abstract

We study the threshold theta geq 2 contact process on a homogeneous tree T_b of degree kappa = b + 1, with infection parameter lambda geq 0 and started from a product measure with density p. The corresponding mean-field model displays a discontinuous transition at a critical point lambda_c^MF(kappa,theta) and for lambda geq lambda_c^MF(kappa,theta) it survives iff p geq p_c^MF(kappa,theta,lambda), where this critical density satisfies 0 < p_c^MF(kappa,theta,lambda) < 1, lim_{lambda to infty} p_c^MF(kappa,theta,lambda) = 0. For large b, we show that the process on T_b has a qualitatively similar behavior when lambda is small, including the behavior at and close to the critical point lambda_c(T_b,theta). In contrast, for large lambda the behavior of the process on T_b is qualitatively distinct from that of the mean-field model in that the critical density has p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0. We also show that lim_{b to infty} b lambda_c(T_b,theta) = Phi_theta, where 1 < Phi₂ < Phi₃ < ..., lim_{theta to infty} Phi_theta = infty, and 0 < liminf_{b to infty} b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty} b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty.

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