Multirange percolation of words on the hypercubic lattice

par Roger Silva (Universidade Federal de Minas Gerais)

jeudi 3 juil. 2025, 14:00 → 15:00 Europe/Paris

125 (Université Lyon 1)

We investigate the problem of percolation of words in a random environment. We independently assign each vertex a letter $0$ or $1$ according to Bernoulli random variables with mean $p$. The environment is the resulting graph obtained from an independent long-range bond percolation configuration on ${\mathbb{Z}}^{d-1}\times \mathbb{Z}$, $d\ge 3$, where each edge parallel to ${\mathbb{Z}}^{d-1}$ has length one and is open with probability $ϵ$, while edges of length $n$ parallel to $\mathbb{Z}$ are open with probability $p_n$. We prove that if the sum of $p_n$ diverges, then for any $ϵ$ and $p$, there is a $K$ such that all words are seen from the origin with probability close to $1$, even if all connections with length larger than $K$ are suppressed.