Phase transition on randomly horizontally stretched square lattice

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 $$\mathbb{E}\left[\xi e^{c(\log \xi {)}^{1\mathrm{/}2}}{\mathbb{1}}_{\{\xi \ge 1\}}\right]<\mathrm{\infty },$$ 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 $p^{\mathrm{\mid }e\mathrm{\mid }}$, where $\mathrm{\mid }e\mathrm{\mid }$ 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 $p<1$.