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The Rightmost Particle of the Contact Process on Dynamic Random Environments

Isabella Alvarenga, Aurelia Deshayes

TL;DR

This work analyzes the rightmost front of two 1D contact-process–type models evolving in a dynamic random environment: the Spont process and the contact process with inherited sterility (IS). It develops a renewal-time approach based on active infection paths to establish a law of large numbers and a central limit theorem for the position of the rightmost fertile site, under survival for each model. The lack of monotonicity in IS (and non-self-duality for Spont) motivates a detailed renewal-structure construction that yields i.i.d. increments and robust tail bounds, enabling precise fluctuation results. The resulting framework advances understanding of front propagation in interacting particle systems with environment–process coupling and suggests pathways for extending to broader dynamic environments and higher dimensions.

Abstract

We study the behaviour of the rightmost occupied site in two models: the Spont process and the contact process with inherited sterility, in dimension 1. Both can be viewed as contact processes evolving in dynamic random environments, where the environment may itself depend on the state of the process. In the Spont process, blocking particles appear spontaneously, while in the inherited sterility model, sterile sites arise as offspring of occupied ones. Each model presents distinct mathematical challenges: the Spont process lacks self-duality, whereas the inherited sterility process is non-attractive. We establish a law of large numbers and a central limit theorem for the position of the rightmost occupied site. Our approach is based on the construction of a sequence of renewal times, defined through a detailed analysis of active infection paths. These results are obtained in the supercritical regime of the Spont process.

The Rightmost Particle of the Contact Process on Dynamic Random Environments

TL;DR

This work analyzes the rightmost front of two 1D contact-process–type models evolving in a dynamic random environment: the Spont process and the contact process with inherited sterility (IS). It develops a renewal-time approach based on active infection paths to establish a law of large numbers and a central limit theorem for the position of the rightmost fertile site, under survival for each model. The lack of monotonicity in IS (and non-self-duality for Spont) motivates a detailed renewal-structure construction that yields i.i.d. increments and robust tail bounds, enabling precise fluctuation results. The resulting framework advances understanding of front propagation in interacting particle systems with environment–process coupling and suggests pathways for extending to broader dynamic environments and higher dimensions.

Abstract

We study the behaviour of the rightmost occupied site in two models: the Spont process and the contact process with inherited sterility, in dimension 1. Both can be viewed as contact processes evolving in dynamic random environments, where the environment may itself depend on the state of the process. In the Spont process, blocking particles appear spontaneously, while in the inherited sterility model, sterile sites arise as offspring of occupied ones. Each model presents distinct mathematical challenges: the Spont process lacks self-duality, whereas the inherited sterility process is non-attractive. We establish a law of large numbers and a central limit theorem for the position of the rightmost occupied site. Our approach is based on the construction of a sequence of renewal times, defined through a detailed analysis of active infection paths. These results are obtained in the supercritical regime of the Spont process.
Paper Structure (21 sections, 22 theorems, 93 equations, 5 figures)

This paper contains 21 sections, 22 theorems, 93 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Models and main results
  3. Notations about configurations
  4. Spont dynamics
  5. IS dynamics
  6. Graphical construction
  7. Attractivity and lack of attractivity
  8. Special property and renewal times
  9. Special property
  10. Definition of renewal times
  11. Estimates on the occurrence of the special property
  12. Conditions for discrepancies in Spont rightmost positions
  13. Proof of Propositions \ref{['FPossibleAndSmallCluster']} and \ref{['Prop_Time_Between_ren']}
  14. Conditions for discrepancies in IS rightmost positions
  15. Proof of Proposition \ref{['PropertiesFailureTimeCPIS']} and \ref{['Prop_Time_Between_renCPIS']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $(\xi_t)$ be a Spont process with parameters $\lambda>\lambda_c$ and $p>\tilde{p}$ as in Remark SupercriticalSpont. Then, there exists a constant $\mu_{\mathrm{SP}}>0$ such that, for any initial configuration $c\in\mathcal{C}$, conditioned on survival, we have that

Figures (5)

  • Figure 1: Simulation of the coupled processes.
  • Figure 2: Illustration of graphical construction
  • Figure 3: Event $\mathcal{P}(\chi_\cdot,c,[t_1,t_2])$ as in Definition \ref{['specialPropertySpont']} in case it is not empty
  • Figure 4: Illustration of argument used in Proposition \ref{['FPossibleAndSmallCluster']}. The rightmost particle stays inside the red region due to event $R$, and all the sites above the green region have seen a healing mark due to event $H$; the beginning of the dynamics is controlled by the good event $I$ that appears in blue, where all the sites have seen a healing mark.
  • Figure 5: Coupled processes. (a)–(b): one-dimensional cases. (c): two-dimensional case.

Theorems & Definitions (64)

  • Remark 1: Theorem 4 of velasco2024
  • Theorem 1: Speed of the rightmost particle for Spont
  • Theorem 2: CLT the rightmost particle for Spont
  • Remark 2
  • Remark 3: Theorem 1 of velasco2024
  • Theorem 3: Speed of the rightmost particle for IS
  • Theorem 4: CLT for the rightmost particle for IS
  • Definition 1
  • Lemma 1
  • Remark 4
  • ...and 54 more