Multirange percolation of words on the hypercubic lattice

by Roger Silva (Universidade Federal de Minas Gerais)

Thursday Jul 3, 2025, 2:00 PM → 3:00 PM 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 $\mathbb Z^{d-1} \times \mathbb Z$, $d\geq 3$, where each edge parallel to $\mathbb Z^{d-1}$ has length one and is open with probability $\epsilon$, 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 $\epsilon$ 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.