  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.
