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arXiv · 2506.17444

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
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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.

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Pablo A. Gomes, Marcelo R. Hilário, Bernardo N. B. de Lima, Thomas Mountford. 2025-06-20. The extinction of the contact process in a one-dimensional random environment with long-range interactions. https://arxiv.org/abs/2506.17444

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