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

Weberson S. Arcanjo, Alan S. Pereira, Diogo C. dos Santos, Roger W. C. Silva, Marco Ticse

TL;DR

This work analyzes the constrained-degree percolation model on the hypercubic lattice, where edges open only when both endpoints have fewer than $k$ incident open edges at the moment of activation. Using the decreasing-cluster framework and a suite of local-boundary modification arguments, the authors prove a sharp uniqueness result: for any fixed $k\le 2d$ and all times $t\in[0,1)$, the number of infinite open clusters is almost surely either $0$ or $1$, even in the presence of long-range dependencies. They further show a uniform positivity of the two-point connectivity in the supercritical regime and establish the continuity of the percolation function $\theta(t)$ for $t>t_c$, together with a time-differentiability result for probabilities of local events. Collectively, these results adapt and extend classical percolation techniques to a highly constrained, non-FKG, dependence-rich setting, yielding a precise description of phase behavior and temporal regularity with potential implications for related kinetically constrained models.

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.

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

TL;DR

This work analyzes the constrained-degree percolation model on the hypercubic lattice, where edges open only when both endpoints have fewer than incident open edges at the moment of activation. Using the decreasing-cluster framework and a suite of local-boundary modification arguments, the authors prove a sharp uniqueness result: for any fixed and all times , the number of infinite open clusters is almost surely either or , even in the presence of long-range dependencies. They further show a uniform positivity of the two-point connectivity in the supercritical regime and establish the continuity of the percolation function for , together with a time-differentiability result for probabilities of local events. Collectively, these results adapt and extend classical percolation techniques to a highly constrained, non-FKG, dependence-rich setting, yielding a precise description of phase behavior and temporal regularity with potential implications for related kinetically constrained models.

Abstract

We consider the constrained-degree percolation (CDP) model on the hypercubic lattice. This is a continuous-time percolation model defined by a sequence of i.i.d. uniform random variables and a positive integer , referred to as the constraint. The model evolves as follows: each edge attempts to open at a random time , independently of all other edges. It succeeds if, at time , both of its end-vertices have degrees strictly smaller than . It is known \cite{hartarsky2022weakly} that this model undergoes a phase transition when for most nontrivial values of . In this work, we prove that, for any fixed constraint, the number of infinite clusters at any time is almost surely either 0 or 1. This uniqueness result implies the continuity of the percolation function in the supercritical regime, , where 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.
Paper Structure (11 sections, 7 theorems, 70 equations, 2 figures)

This paper contains 11 sections, 7 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Results
  4. Notation
  5. The decreasing cluster method
  6. Proof of Theorem \ref{['thm: trifurcação']}
  7. The number of infinite clusters is either 0, 1, or infinity
  8. Impossibility of the existence of an infinite number of infinite clusters
  9. Proof of Theorem \ref{['conn1']}
  10. Differentiability of $\mathbb{P}$ for local events
  11. Proofs of claims

Key Result

Theorem 1

Consider the CDP model on $\mathbb{L}^d$ with constraint $k\leq 2d$. Let $\mathcal{N}_t$ denote the number of infinite clusters at time $t$. Then, for all $t\in[0,1)$, it holds that

Figures (2)

  • Figure 1: Boxes $\Lambda_n$ and $\Lambda_{n-1}$ with the sites $w_i, v_i$, for $i\in \{1,2\}$, highlighted. The paths $\gamma_1$ and $\gamma_2$ connect $v_1$ to $w_1$ and $v_2$ to $w_2$, respectively. We build a path $\alpha$ by modifying the states inside of $\Lambda_n$.
  • Figure 2: $\Lambda_n$ after the modifications that generate the trifurcation point $v$.

Theorems & Definitions (21)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Proposition 2
  • ...and 11 more