In Catalan percolation, all nearest-neighbour edges \(\{i,i+1\}\) along \({\mathbb {Z}}\) are initially occupied, and all other edges are open independently with probability p. Open edges \(\{i,j\}\) are occupied if some pair of edges \(\{i,k\}\) and \(\{k,j\}\) , with \(i<k<j\) , become occupied. This model was introduced by Gravner and the third author, in the context of polluted graph bootstrap percolation. We prove that the critical \(p_{\mathrm c}\) is strictly between that of oriented site percolation on \({\mathbb {Z}}^2\) and the Catalan growth rate 1/4. Our main result shows that an enhanced oriented percolation model, with non-decaying, infinite-range dependency, has a strictly smaller critical parameter than the classical model. This is reminiscent of the work of Duminil-Copin, Hilário, Kozma and Sidoravicius on brochette percolation. Our proof differs, however, in that we do not use Aizenman–Grimmett enhancements or differential inequalities. Two key ingredients are the work of Hilário, Sá, Sanchis and Teixeira on stretched lattices, and the Russo–Seymour–Welsh result for oriented percolation by Duminil-Copin, Tassion and Teixeira.
