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

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 T n c → 1 and 2dλ Z d c → 1 as n, d → +∞. Our result is a development of the conclusion that λ Z d c ≤ 2.18 d shown in [2]. 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 (d) n = 0 = 1, where S (d) n is a simple random walk on Z d with S (d) 0 = 0.

Convergence Rates for Subcritical Threshold-One Contact Processes on Lattices

“…According to this dual relationship, [2] shows that the critical value λ c (d) for the threshold-one contact process on Z d satisfies λ c (d) ≤ 2.18/d. [11] develops this result by showing that lim d→+∞ 2dλ c (d) = 1. In recent years, there are some works concerned with threshold contact processes with threshold K > 1.…”

“…The proofs of Theorem 2.2 for cases where λ = 0 and λ > 1/2 are trivial. According to [11], lim d→+∞ 2dλ c (d) = 1.…”

“…In this section we will give lower bounds for I(λ/d, d) and J(λ/d, d) for λ ∈ (0, 1/2). The main tool we utilize is a Markov process {ζ t } t≥0 with state space [0, +∞) Z d introduced in [11]. In other words, for {ζ t } t≥0 , there is a spin at each vertex of Z d taking a nonnegative value.…”

On some threshold-one attractive interacting particle systems on homogeneous trees

“…More recently, in [23], Xue revisits this case and calls it the 'threshold-one contact process'. In [23], Xue shows that the critical birth/infection rate λ for the threshold-one contact process on a high-dimensional lattice is ∼ (2d) −1 , where d is the dimension. Later, in [24], he shows the convergence rate when the system dies out.…”

“…In [5] and [21], the process restricted on a finite subtree is also discussed. When θ = 1, for the threshold-one contact process, [23] shows that the critical value is ∼d −1 for large d. However, until now, little is known, to our knowledge, about the threshold voter model on T d .…”