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The extinction of the contact process in a one-dimensional random environment with long-range interactions

Pablo A. Gomes, Marcelo R. Hilário, Bernardo N. B. de Lima, Thomas Mountford

Abstract

We study the contact process on the long-range percolation cluster on $\mathbb{Z}$ where each edge $\langle i,j \rangle$ is open with probability $|i-j|^{-s}$ for $s> 2$. Using a renormalization procedure we apply Peierls-type argument to prove that the contact process dies out if the transmission rate is smaller than a critical threshold. Our methods involve the control of crossing probabilities for percolation on randomly-stretched lattices as in https://doi.org/10.1214/22-AAP1887.

The extinction of the contact process in a one-dimensional random environment with long-range interactions

Abstract

We study the contact process on the long-range percolation cluster on where each edge is open with probability for . Using a renormalization procedure we apply Peierls-type argument to prove that the contact process dies out if the transmission rate is smaller than a critical threshold. Our methods involve the control of crossing probabilities for percolation on randomly-stretched lattices as in https://doi.org/10.1214/22-AAP1887.

Paper Structure

This paper contains 10 sections, 14 theorems, 99 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Model and definitions
  4. Related works
  5. Summary of the paper
  6. Control of the environment, cut-points
  7. The renormalization step
  8. Crossing events
  9. Percolation on stretched lattices
  10. Proof of Theorem \ref{['theo1']}

Key Result

Theorem 1

If $s>2$ then the critical infection rate for the contact process on the graph $G=(\mathbb{Z},\mathcal{E})$ satisfies $\lambda_c >0$.

Figures (9)

  • Figure 1: A realization of the long-range percolation. Assumption $s>2$ guarantees existence of infinitely many cut-points (black and red). Strong cut-points (red) are surrounded by cut-points.
  • Figure 2: Graphical representation of the contact process on a random one-dimensional graph with long-range connections. Lozenges (blue) represent recovery marks, while horizontal double arrows (gray and red) indicate potential transmission marks. The red path illustrates the spread of infection starting from the vertex $o$.
  • Figure 3: The representation of the strong cut-points $v_k$ and the corresponding sets $V_{2k}$.
  • Figure 4: The renormalization scheme. We represent the event $\{(2k,1) \text{ is good}\}$. In the corresponding rectangle $[\space [ v_{k}+1, v_{k+1}-1 ] \space] \times [T,2T]$ every site has a recovery mark (lozenge) and no edge has an infection attempt (double arrow).
  • Figure 5: The lattice $\mathcal{L}_\Lambda$ (on the left) and $\mathbb{Z}\times \mathbb{Z}_+$ (on the right). The environment $\Lambda$ can be specified either by the $x_k$'s (left) or by the $\xi_k=x_{k}-x_{k-1}$ (right).
  • ...and 4 more figures

Theorems & Definitions (31)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 2
  • Lemma 4
  • Remark 3
  • Lemma 5
  • Lemma 6
  • ...and 21 more