On the number of infinite clusters in the constrained-degree percolation model

We consider constrained-degree percolation model on the hypercubic lattice Lᵈ= (Zᵈ, Eᵈ). In this model, there exits a sequence (Uₑ) ₄㵧 of i. i. d. random variables with distribution Unif (0, 1) and a positive integer k, which is called a constraint. Each edge e attempts to open at time Uₑ, and the attempt is successful if the number of neighboring edges open at each endvertex of e is at most k-1. In hartarsky2022weakly, the authors demonstrated that this model undergoes a phase transition when d3 and for most nontrivial values of k. In the present work, we prove that, for any fixed constraint, the number of infinite clusters at any given time t[0, 1) is either 0 or 1, almost surely.