We consider constrained-degree percolation on the hypercubic lattice. This is a continuous-time model defined by a sequence \((U_e)_{e}\) of i.i.d. uniform random variables and a positive integer k, referred to as the constraint. The model evolves as follows: each edge e attempts to open at a random time \(U_e\) , independently of all other edges. It succeeds if, at time \(U_e\) , both of its end-vertices have degrees strictly smaller than k. It is known [21] that this model undergoes a phase transition when \(d\ge 3\) for most nontrivial values of k. In this work, we prove that, for any fixed constraint, the number of infinite clusters at any time \(t\in [0,1)\) is almost surely either 0 or 1. We also show that the law of the process is differentiable with respect to time for local events, extending a result of [30]. As a consequence of these two results, we prove that the percolation function is continuous in the supercritical regime \(t\in (t_c,1)\) , where \(t_c\) denotes the percolation critical threshold. Finally, we show that the two-point connectivity function is bounded away from zero in the supercritical regime.
