Truncated long-range percolation of words on the square lattice

2024/12/26 by Pablo A. Gomes, Pablo Almeida Gomes, Otávio Lima +5
Mathematics · #Advanced Combinatorial Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.2412.19379

openalex publication_date 2024/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We study mixed long-range percolation on the square lattice. Each vertical edge of unit length is independently open with probability ε, and each horizontal edge of length i is independently open with probability p_i. Also, each vertex is assigned independently a random variable taking values 1 and 0 with probability p and 1-p, respectively. We prove that for a broad class of anisotropic long-range percolation models for which connection probabilities p_i satisfy some regularity conditions, all words (semi-infinite binary sequences) are seen simultaneously from the origin with positive probability, even if all edges with length larger than some constant (depending on ε, p, and on the sequence (p_i)) are suppressed.

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