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 Z₊² according to a collection of independent and identically distributed (i.i.d.) copies of a non-negative random variable $ξ$. We assume that $ξ$ satisfies the integrability condition \[ E[ξ\, e^{c(log ξ)1/2} \,1_{\{ξ≥ 1\}}] < ∞, \] for some constant c > 8√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 $p < 1$.