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)eᵈ 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 d≥3 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.