Exponential decay for Constrained-degree percolation

TL;DR

This paper proves exponential decay of the open cluster radius in a constrained-degree percolation model on a lattice, establishing a phase transition and conditions for finite expected cluster size.

Contribution

It introduces a new constrained percolation model in a random environment and proves exponential decay and a phase transition for the cluster size.

Findings

Exponential decay of cluster radius when expected size is finite

Existence of a sharp phase transition in the model

Finite expected cluster size for certain time intervals

Abstract

We consider the Constrained-degree percolation model in random environment (CDPRE) 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 $\rho_j$. The dynamics is as follows: at time $t=0$ all edges are closed; each edge $e$ attempts to open at a random time $U_e\sim \mathrm{U}(0,1]$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respective constraints. We obtain exponential decay of the radius of the open cluster of the origin at all times when its expected size is finite. Since CDPRE is dominated by Bernoulli percolation, such result is meaningful only if the supremum of all values of $t$ for which the expected size of the open cluster of the origin is finite is larger than 1/2. We prove this last fact by showing a sharp phase transition for an intermediate model.