Truncated long-range percolation of words on the square lattice

We study mixed long-range percolation on the square lattice. Each vertical edge of unit length is independently open with probability $\varepsilon$, 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 $\varepsilon$, $p$, and on the sequence $(p_i)$) are suppressed.