Constrained-degree percolation in random environment

Abstract

We consider the Constrained-degree percolation model with random constraints on the square lattice and prove a non-trivial phase transition. In this model, each vertex has an independently distributed random constraint $j\in \{0,1,2,3\}$ with probability $\rho_j$. Each edge $e$ tries to open at a random uniform time $U_e$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respectively attached constraints. We show that this model undergoes a non-trivial phase transition when $\rho_3$ is sufficiently large. The proof consists of a decoupling inequality, the continuity of the probability for local events, together with a coarse-graining argument.