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Limiting Speed and Fluctuations for the Boundary Modified Contact Process

Andrew Heeszel

TL;DR

This work analyzes the one-dimensional boundary modified contact process in the critical-right-edge regime with a positive edge boost, establishing a strong law of large numbers and a central limit theorem for the rightmost infected site. The authors introduce a right-edge auxiliary process sampled from an invariant measure to characterize the asymptotic speed $\alpha=\mathbb{E}(\mathcal{R}(\tilde{\eta}_1))$, show $\mathcal{R}(\eta_t)/t\to\alpha$ a.s., and prove a diffusion limit under proper scaling. They develop stretched-exponential tail bounds for extinction and survival events, and leverage box-crossing properties of critical oriented percolation to handle non-attractiveness and loss of subadditivity. The results extend to the boundary-modified model and resolve an open question raised by Andjel and Rolla, providing a rigorous probabilistic description of edge fluctuations via an $\alpha$-mixing functional CLT with a strictly positive diffusion coefficient. This advances understanding of edge dynamics in nonuniform contact processes and connects to percolation-box-crossing techniques in low dimensions.

Abstract

The boundary modified contact process models an epidemic spreading in one dimension with two infection parameters, $λ_i$ and $λ_e$. Starting from a finite infected set, each edge of $\mathbb{Z}$ transmits the infection at rate $λ_i$ except for the rightmost and leftmost edges incident to infected vertices, which transmit the infection at rate $λ_e$. We show a strong law of large numbers and central limit theorem for the location of the rightmost infected vertex when $λ_i = λ_c$ and $λ_e = λ_c + \varepsilon$. We also show stretched exponential tail bounds in the fluctuations of the rightmost infected vertex, the extinction time of the process on the event of non-survival, and the probability of survival given the size of the initial infected region. Our results extend to the boundary modified contact process whenever $λ_c \leq λ_i < λ_e$, and solves an open problem first proposed by Andjel and Rolla in [1].

Limiting Speed and Fluctuations for the Boundary Modified Contact Process

TL;DR

This work analyzes the one-dimensional boundary modified contact process in the critical-right-edge regime with a positive edge boost, establishing a strong law of large numbers and a central limit theorem for the rightmost infected site. The authors introduce a right-edge auxiliary process sampled from an invariant measure to characterize the asymptotic speed , show a.s., and prove a diffusion limit under proper scaling. They develop stretched-exponential tail bounds for extinction and survival events, and leverage box-crossing properties of critical oriented percolation to handle non-attractiveness and loss of subadditivity. The results extend to the boundary-modified model and resolve an open question raised by Andjel and Rolla, providing a rigorous probabilistic description of edge fluctuations via an -mixing functional CLT with a strictly positive diffusion coefficient. This advances understanding of edge dynamics in nonuniform contact processes and connects to percolation-box-crossing techniques in low dimensions.

Abstract

The boundary modified contact process models an epidemic spreading in one dimension with two infection parameters, and . Starting from a finite infected set, each edge of transmits the infection at rate except for the rightmost and leftmost edges incident to infected vertices, which transmit the infection at rate . We show a strong law of large numbers and central limit theorem for the location of the rightmost infected vertex when and . We also show stretched exponential tail bounds in the fluctuations of the rightmost infected vertex, the extinction time of the process on the event of non-survival, and the probability of survival given the size of the initial infected region. Our results extend to the boundary modified contact process whenever , and solves an open problem first proposed by Andjel and Rolla in [1].

Paper Structure

This paper contains 19 sections, 20 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notation and a Note on Constants
  4. Sitewise Construction & Attractiveness at the Right Edge
  5. Bounding Survival Function of $\eta(t)$ on Extinction
  6. Bounding forward increments of $\mathcal{R}(\eta'_t)$ on event of survival by time $t$
  7. Forming a High Probability Survival Event
  8. Proof of Theorems 3 & 4
  9. Law of Large Numbers for $\mathcal{R}(\eta_t)$
  10. Bounding Increments of $\mathcal{R}(\eta_t)$ under $\tilde{\mu}$
  11. Large Deviations for $\mathcal{R}(\tilde{\eta}_t)$
  12. Strong Law of Large Numbers
  13. Lower Bound to Edge Speeds
  14. Brownian Motion Approximation
  15. Setup
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\xi_t$ be either the boundary modified or right edge modified contact process with infection rates $\lambda_i = \lambda_c$ and $\lambda_e = \lambda_c + \varepsilon$ for some $\varepsilon > 0$. Then when $|\xi_0| < \infty$,

Figures (1)

  • Figure 1: Figures of the event $F_{k}$ and $G_{k}$ occurring on the left, and the right edge having linear speed during an epoch $F_{ik}$ on the right. Paths open under the critical contact process are in blue, while $\lambda_e$-open paths are in red.

Theorems & Definitions (37)

  • Lemma 1: Andjel, Rolla
  • Lemma 2: Duminil-Copin, Tassion, Teixeira
  • Lemma 3: Duminil-Copin, Tassion, Teixeira
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 1
  • Lemma 4
  • ...and 27 more