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 (ξ) ^{1/2} \, 1\⏝ ₁\ 8 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.