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

Abstract We consider constrained-degree percolation on the hypercubic lattice. This is a continuous-time model defined by a sequence (Uₑ) ₄ (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.