Critical value for the contact process with random recovery rates and edge weights on regular tree.pdfVIP

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Critical value for the contact process with random recovery rates and edge weights on regular tree.pdf

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Critical value for the contact process with random recovery rates and edge weights on regular tree.pdf

Physica A 462 (2016) 793–806 Contents lists available at ScienceDirect Physica A journal homepage: /locate/physa Critical value for the contact process with random recovery rates and edge weights on regular tree Xiaofeng Xue School of Science, Beijing Jiaotong University, Beijing 100044, China highlights ? We study contact process with random recovery rates ξ and edge weights ρ on tree TN . ? We introduce the definitions of the critical values under the annealed and quenched measures. ? We show that the critical value under the annealed measure asymptotically equals (N Eρ E 1 ξ )?1 as N grows to infinity. ? We show that the critical value under the quenched measure equals that under the annealed measure on a set A and equals infinity on the complement of A. article info Article history: Received 30 March 2016 Received in revised form 2 June 2016 Available online 22 June 2016 Keywords: Contact process Regular tree Edge weight Recovery rate Critical value abstract In this paper we are concerned with contact processes with random recovery rates and edge weights on rooted regular trees TN . Let ρ and ξ be two nonnegative random variables such that P(? ≤ ξ +∞, ρ ≤ M) = 1 for some ?, M 0. For each vertex x on TN , ξ (x) is an independent copy of ξ while for each edge e on TN , ρ(e) is an independent copy of ρ. An infected vertex x becomes healthy at rate ξ (x) while an infected vertex y infects an healthy neighbor z at rate proportional to ρ(y, z). For this model, we prove that the critical value under the annealed measure approximately equals (N Eρ E 1 ξ )?1 as N grows to infinity. Furthermore, we show that the critical value under the quenched measure equals that under the annealed measure when the cluster containing the root formed with edges with positive weights is infinite. ? 2016 Published by Elsevier B.V. 1. Introduction In this paper, we are concerned about contact processes with random recovery rates and edge weights on regular trees. For each integer N ≥ 1, w