Phase Transition for the Large-Dimensional Contact Process with Random Recovery Rates on Open Clusters

Abstract

In this paper we are concerned with contact process with random recovery rates on open clusters of bond percolation on Z d . Let ξ be a positive random variable, then we assigned i. i. d. copies of ξ on the vertices as the random recovery rates. Assuming that each edge is open with probability p and log d vertices are occupied at t = 0, we prove that the following phase transition occurs. When the infection rate λ < λ c = 1/(pE 1 ξ ), then the process dies out at time O(log d) with high probability as d → +∞, while when λ > λ c , the process survives with high probability.for each e ∈ E d , where p ∈ (0, 1), then we denote by x ↔ y when and only when x ∼ y and the edge e connecting x and y satisfies that X(e) = 1. For each x ∈ Z d , we defineas the set of neighbors of x. Please note that N (x) is a random set depending on {X(e)} e∈E d . Let ξ be a random variable such that P (ξ ≥ 1) = 1 and {ξ(x)} x∈Z d be i. i. d. random variables such that ξ(x) and ξ have identical probability distributions. We assume that {ξ(x)} x∈Z d and {X(e)} e∈E d are independent. The contact process is a Markov process with state space

Anisotropic oriented percolation in high dimensions

“…In the particular case of oriented percolation, the seminal paper by Cox and Durrett (1983) shows, among other important results, that the asymptotic behavior of the critical parameter is 1/d. A little earlier, Holley and Liggett (1981) proved the occurrence of an analogous behavior for the high dimensional contact process critical rate and, recently, a similar result was proven for the contact process with random rates on a high dimensional percolation open cluster, see Xue (2016). For the non-oriented case, Kesten (1990) and Gordon (1991) independently showed that the critical parameter is asymptotically 1/2d and, in the last three decades, a rather complete mean-field picture of high dimensional non-oriented percolation has emerged; see Heydenreich and van der Hofstad (2017) and references therein.…”

Anisotropic oriented percolation in high dimensions

“…In the particular case of oriented percolation, the seminal paper by Cox and Durrett [1] shows, among other important results, that the asymptotic behavior of the critical point is 1/d. A little earlier, Holley and Liggett [4] proved the occurrence of an analogous behavior for the high dimensional contact process critical rate and, recently, a similar result was prove for the contact process with random rates on a high dimentional percolation open cluster, see [8]. For the non-oriented case, Kesten [5] and Gordon [3] independently showed that the critical point is asymptotically 1/2d and, in the last three decades, a rather complete mean-field picture of high dimensional non-oriented percolation has emerged; see [6] and references therein.…”

Asymptotic of the Critical Value of the Large-Dimensional SIR Epidemic on Clusters

“…This result has been proved in [12] in a general case where each infective vertex recovers at i. i. d. random rates. We believe that lim inf…”

“…Let λ c (d) be the counterpart of λ c (d) with respect to the contact process, then it is proved in [4] that lim This result has been proved in [12] in a general case where each infective vertex recovers at i. i. d. random rates. We believe that lim inf…”