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

We consider constrained-degree percolation model on the hypercubic lattice \linebreak $\mathbb{L}^d=(\mathbb{Z}^d,\mathbb{E}^d)$. In this model, there exits a sequence $(U_e)_{e\in\mathbb{E}^d}$ 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 $U_e$, and the attempt is successful if the number of neighboring edges open at each endvertex of $e$ is at most $k-1$. In \cite{hartarsky2022weakly}, the authors demonstrated that this model undergoes a phase transition when $d\geq3$ 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 $t\in[0,1)$ is either 0 or 1, almost surely.
Comment: 15 pages