Exponential decay for Constrained-degree percolation

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.
Comment: 15 pages, 1 figure