Phase transition on randomly horizontally stretched square lattice

Abstract

In this article, we study a bond percolation model on a horizontally stretched square lattice, constructed by stretching the distances between the columns of $\mathbb{Z}_+^2$ according to a collection of independent and identically distributed (i.i.d.) copies of a non-negative random variable $\xi$. We assume that $\xi$ satisfies the integrability condition \[ \mathbb{E}\big[\xi\, e^{c(\log \xi)^{1/2}} \,\mathbb{1}_{\{\xi \geq 1\}}\big] < \infty, \] for some constant $c > 8\sqrt{\log 96}$. In this random environment, each vertical edge is independently declared open with probability $p$, while each horizontal edge is open with probability $p^{|e|}$, where $|e|$ denotes the Euclidean length of the edge. We develop a multiscale renormalization scheme adapted to this geometry and use it to prove that percolation occurs for all sufficiently large values of $p < 1$.