Constrained-degree percolation in random environment

TL;DR

This paper studies a percolation model on a lattice where vertices have random constraints affecting edge opening, revealing a phase transition influenced by the distribution of these constraints.

Contribution

It introduces and analyzes a constrained-degree percolation model in a random environment, proving a phase transition based on the distribution of vertex constraints.

Findings

Model undergoes a non-trivial phase transition when the probability of maximum constraint is high.

Established a decoupling inequality and continuity of local event probabilities.

Used coarse-graining techniques to analyze the phase transition.

Abstract

We consider the Constrained-degree percolation model in random environment on the square lattice. In this model, each vertex $v$ has an independent random constraint ${\kappa}_v$ which takes the value $j\in \{0,1,2,3\}$ with probability $\rho_j$. Each edge $e$ attempts to open at a random uniform time $U_e$ in $[0,1]$, 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, and a coarse-graining argument.