Constrained-Degree Percolation on the Hypercubic Lattice: Uniqueness and Some of its Consequences

We consider constrained-degree percolation on the hypercubic lattice. This is a continuous-time model defined by a sequence (Uₑ)ₑ ( 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 Uₑ U e , independently of all other edges. It succeeds if, at time Uₑ 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≥ 3 d ≥ 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∈ [0,1) t ∈ [ 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∈ (tc,1) t ∈ ( t c , 1 ) , where tc 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.