Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Contact Process Can Survive on a Slightly Subcritical Dynamical Percolation Cluster

Aurelia Deshayes, Régine Marchand

TL;DR

The paper studies the contact process on a dynamically evolving graph (CPDE) on $\mathbb{Z}^d$, where edges switch between available and unavailable states with rates $vp$ and $v(1-p)$. It proves that for every dimension $d\ge 2$ there exists a slightly subcritical edge density $p<p_c(d)$ such that, for all update speeds $v>0$, the infection can survive provided the infection rate $\lambda$ is large enough. The authors introduce an algorithmic framework that incrementally reveals an infected cluster and couples it to a supercritical percolation, circumventing earlier block-based methods. A key novelty is the extension to $p<p_c(d)$ via second-chance infections on a subset of edges, leading to a two-type percolation/enhancement argument that yields survival in the subcritical regime. The results demonstrate that dynamic environments can sustain infection even below the static percolation threshold, enriching the understanding of phase transitions in CPDE models and extending the one-dimensional findings to higher dimensions.

Abstract

The contact process on dynamic edges (CPDE) is a contact process evolving on a dynamic environment given by a dynamical percolation on the edges of Z d\,: each edge updates its state to open or closed with respective rates vp and v(1 -p). By coupling a well-chosen subset of once infected sites in the CPDE with a cluster of some supercritical percolation on the edges of Z d , we prove that, for every dimension d $\ge$ 2, we can find some slightly subcritical p < pc(d) such that for every update speed v > 0, the contact process with large enough infection rate can survive. This extends the result for dimension 1 proved by Linker and Remenik in [LR20].

The Contact Process Can Survive on a Slightly Subcritical Dynamical Percolation Cluster

TL;DR

The paper studies the contact process on a dynamically evolving graph (CPDE) on , where edges switch between available and unavailable states with rates and . It proves that for every dimension there exists a slightly subcritical edge density such that, for all update speeds , the infection can survive provided the infection rate is large enough. The authors introduce an algorithmic framework that incrementally reveals an infected cluster and couples it to a supercritical percolation, circumventing earlier block-based methods. A key novelty is the extension to via second-chance infections on a subset of edges, leading to a two-type percolation/enhancement argument that yields survival in the subcritical regime. The results demonstrate that dynamic environments can sustain infection even below the static percolation threshold, enriching the understanding of phase transitions in CPDE models and extending the one-dimensional findings to higher dimensions.

Abstract

The contact process on dynamic edges (CPDE) is a contact process evolving on a dynamic environment given by a dynamical percolation on the edges of Z d\,: each edge updates its state to open or closed with respective rates vp and v(1 -p). By coupling a well-chosen subset of once infected sites in the CPDE with a cluster of some supercritical percolation on the edges of Z d , we prove that, for every dimension d 2, we can find some slightly subcritical p < pc(d) such that for every update speed v > 0, the contact process with large enough infection rate can survive. This extends the result for dimension 1 proved by Linker and Remenik in [LR20].
Paper Structure (7 sections, 3 theorems, 41 equations, 1 figure, 4 algorithms)

This paper contains 7 sections, 3 theorems, 41 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Model and results.
  3. Model
  4. Previous results
  5. Result and sketch of proof
  6. Warm up: the supercritical case $p>p_c(d)$
  7. Slightly subcritical case, with second chance infections

Key Result

Theorem 2.1

Let $d \ge 2$ be fixed. Then, In other words, $p_1 < p_c(d)$.

Figures (1)

  • Figure 1: The first chance infection through $(x,y)$, encoded in $\xi_{(x,y)}$, succeeds, then the second chance infection through $(y,z)$ with the help of $y$'s neighbor $x$, encoded by $\zeta^x_{(y,z)}$, succeeds thanks to a type $(\star\star)$ event. The time slots when edges are available are in pink. The contact process $\tilde{\eta}_{\{x,y\}}$ is simulated during the darker slot.

Theorems & Definitions (5)

  • Theorem 2.1
  • Lemma 4.1
  • proof
  • Definition 4.2
  • Lemma 4.3: Aizenman-Grimmett