  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.
