Two Phase Transitions for the Contact Process on Small Worlds

Sharpness versus robustness of the percolation transition in 2D contact processes

We study versions of the contact process with three states, and with infections occurring at a rate depending on the overall infection density. Motivated by a model described in [17] for vegetation patterns in arid landscapes, we focus on percolation under invariant measures of such processes. We prove that the percolation transition is sharp (for one of our models this requires a reasonable assumption). This is shown to contradict a form of 'robust critical behaviour' with power law cluster size distribution for a range of parameter values, as suggested in [17].

Sharpness of the Percolation Phase Transition for the Contact Process on $\mathbb{Z}^d$

We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^2$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raou…

Phase transition for large dimensional contact process with random recovery rates on open clusters

In this paper we are concerned with contact process with random recovery rates on open clusters of bond percolation on $\mathbb{Z}^d$. 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 $λ<λ_c=1/({p{\rm E}\frac{1}ξ})$, then the process dies out at time $O(\log d)$ with high probability as $d\rightarrow+\infty$, while when $λ>λ_c$, the process surviv…

Percolation by cumulative merging and phase transition for the contact process on random graphs

Given a weighted graph, we introduce a partition of its vertex set such that the distance between any two clusters is bounded from below by a power of the minimum weight of both clusters. This partition is obtained by recursively merging smaller clusters and cumulating their weights. For several classical random weighted graphs, we show that there exists a phase transition regarding the existence of an infinite cluster. The motivation for introducing this partition arises from a connection with the contact process as it roughly describes the geometry of the sets where the process survives for …

再生接触过程：相变与生存

Renewal Contact Processes: phase transition and survival

本文研究了再生接触过程的相变与生存问题，通过扩展间隔时间分布的条件，证明了临界值的严格正性，并探讨了在生存条件下接触过程的渐近预测问题。

A discontinuous phase transition in the threshold-$θ\geq 2$ contact process on random graphs

We study the discrete-time threshold-$θ\geq 2$ contact process on random graphs of general degrees. For random graphs with a given degree distribution $μ$, we show that if $μ$ is lower bounded by $θ+2$ and has finite $k$th moments for all $k>0$, then the discrete-time threshold-$θ$ contact process on the random graph exhibits a discontinuous phase transition in the emergence of metastability, thus answering a question of Chatterjee and Durrett \cite{cd13}. To be specific, we establish that (i) for any large enough infection probability $p>p_1$, the process started from the all-infected state w…

The Symbiotic Contact Process

We consider a contact process on $Z^d$ with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species $A$ and/or $B$. Multiple occupancy by the same species at a single site is prohibited. The name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $μ\le 1$ for each particle at that site. We show the critical birth rate $λ_c(μ)$ for weak survival is of order $\sqrtμ$ as $μ\to 0$. Mean-field calculations predic…

Phase transitions for a planar quadratic contact process

We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results that give bounds on the critical values $λ_f$ for the process to survive from a finite set and $λ_e$ for the existence of a nontrivial stationary distribution. Our first result comes from a standard block construction, while the second involves a comparison with the "generic population model" of Bramson and Gray (1991). An interesting new feature of our work is the suggestion that, as in the one dimensional contact process, edge speeds characterize critical values. We are able to prove the …