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Exponential decay for Constrained-degree percolation

Diogo C. dos Santos, Roger W. C. Silva

TL;DR

This paper analyzes the Constrained-degree percolation in random environment (CDPRE) on the square lattice, showing that the radius of the open cluster decays exponentially whenever the mean cluster size is finite, and that the susceptibility threshold satisfies $\bar{t}_c(\\rho) > \tfrac{1}{2}$. The authors prove exponential decay below $\bar{t}_c(\\rho)$ via a Simon-Lieb type inequality, and establish $\bar{t}_c(\\rho) > \tfrac{1}{2}$ by introducing an intermediate model whose phase transition is shown to be sharp through OSSS-type arguments and Russo-type formulas. For the case $\\rho_0=0$, they construct a specific 5-dependent intermediate model on boxes and prove its sharpness, while for $\\rho_0>0$ they use an essential diminishment argument to relate CDPRE to Bernoulli percolation. Overall, the work advances understanding of dependent percolation with random constraints, connecting subcritical decay to sharp threshold phenomena and providing a framework for further exploration of equality of critical thresholds in CDPRE.

Abstract

We consider the Constrained-degree percolation model in random environment (CDPRE) on the square lattice. In this model, each vertex $v$ has an independent random constraint $κ_v$ which takes the value $j\in \{0,1,2,3\}$ with probability $ρ_j$. The dynamics is as follows: at time $t=0$ all edges are closed; each edge $e$ attempts to open at a random time $U_e\sim \mathrm{U}(0,1]$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respective constraints. We obtain exponential decay of the radius of the open cluster of the origin at all times when its expected size is finite. Since CDPRE is dominated by Bernoulli percolation, such result is meaningful only if the supremum of all values of $t$ for which the expected size of the open cluster of the origin is finite is larger than 1/2. We prove this last fact by showing a sharp phase transition for an intermediate model.

Exponential decay for Constrained-degree percolation

TL;DR

This paper analyzes the Constrained-degree percolation in random environment (CDPRE) on the square lattice, showing that the radius of the open cluster decays exponentially whenever the mean cluster size is finite, and that the susceptibility threshold satisfies . The authors prove exponential decay below via a Simon-Lieb type inequality, and establish by introducing an intermediate model whose phase transition is shown to be sharp through OSSS-type arguments and Russo-type formulas. For the case , they construct a specific 5-dependent intermediate model on boxes and prove its sharpness, while for they use an essential diminishment argument to relate CDPRE to Bernoulli percolation. Overall, the work advances understanding of dependent percolation with random constraints, connecting subcritical decay to sharp threshold phenomena and providing a framework for further exploration of equality of critical thresholds in CDPRE.

Abstract

We consider the Constrained-degree percolation model in random environment (CDPRE) on the square lattice. In this model, each vertex has an independent random constraint which takes the value with probability . The dynamics is as follows: at time all edges are closed; each edge attempts to open at a random time , independently of all other edges. It succeeds if at time both its end-vertices have degrees strictly smaller than their respective constraints. We obtain exponential decay of the radius of the open cluster of the origin at all times when its expected size is finite. Since CDPRE is dominated by Bernoulli percolation, such result is meaningful only if the supremum of all values of for which the expected size of the open cluster of the origin is finite is larger than 1/2. We prove this last fact by showing a sharp phase transition for an intermediate model.

Paper Structure

This paper contains 9 sections, 6 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results and discussion
  3. Proofs
  4. Proof of Theorem \ref{['conn2']}
  5. Proof of Theorem \ref{['susce']}
  6. Bound on the revealment
  7. A Russo's type formula
  8. A bound on the influences
  9. Final remarks

Key Result

Theorem 1

It holds that $\bar{t}_c(\rho)>\frac{1}{2}$.

Figures (1)

  • Figure 1: $\Lambda_{r,s}$ (larger box), $\overline{\Lambda}_{r,s}$ (gray box) and the edge $g_{r,s}$. $A_{r,s}$ consists of the dashed edges.

Theorems & Definitions (12)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Remark 2
  • Lemma 1
  • Definition 1: Algorithm $\mathbf{T}_k$
  • Lemma 2
  • proof
  • ...and 2 more