Constrained-degree percolation on the hypercubic lattice: uniqueness and some of its consequences

TL;DR

This paper proves the almost sure uniqueness of the infinite cluster in the constrained-degree percolation model on hypercubic lattices, establishing continuity and differentiability of the percolation function over time.

Contribution

It demonstrates that for any fixed constraint, the number of infinite clusters is almost surely 0 or 1, and extends time-regularity results to the CDP model.

Findings

Almost sure uniqueness of infinite clusters for any fixed constraint.

Continuity of the percolation function in the supercritical regime.

Differentiability of the process law with respect to time.

Abstract

We consider the constrained-degree percolation (CDP) model on the hypercubic lattice. This is a continuous-time percolation model defined by a sequence $(U_e)_{e\in\mathcal{E}^d}$ 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 $U_e$, independently of all other edges. It succeeds if, at time $U_e$, both of its end-vertices have degrees strictly smaller than $k$. It is known \cite{hartarsky2022weakly} that this model undergoes a phase transition when $d\geq3$ for most nontrivial values of $k$. In this work, we prove that, for any fixed constraint, the number of infinite clusters at any time $t\in[0,1)$ is almost surely either 0 or 1. This uniqueness result implies the continuity of the percolation function in the supercritical regime, $t\in(t_c,1)$, where $t_c$ denotes the percolation critical threshold. The proof relies on a key time-regularity property of the model: the law of the process is continuous with respect to time for local events. In fact, we establish differentiability in time, thereby extending the result of \cite{SSS} to the CDP setting.