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Critical percolation on slabs with random columnar disorder

Matheus B. Castro, Rémy Sanchis, Roger W. C. Silva

TL;DR

This work analyzes bond percolation on a 3D slab with columnar disorder driven by a heavy-tailed renewal process. The authors develop a multiscale renormalization tailored to heavy-tailed environments, proving that when the enhanced-column probability $q$ is supercritical and the renewal tail exponent $\\phi$ is large enough (depending on the slab thickness $k$), percolation occurs at a horizontal bond probability $p$ strictly below the critical threshold $p_c(\\mathbb{S}^+_k)$. A key contribution is a polynomial upper bound on the slab correlation length $L_\\tau(p)$ in terms of $|p-p_c|$, achieved via a combination of pivotal-edge analysis, Gluing Lemmas, and a 1-dependent renormalized percolation coupled to a Marcos-style multiscale scheme. The results illuminate near-critical behavior in disordered 3D systems with infinite-range vertical dependencies and provide a framework for extending near-critical percolation methods to slab geometries with heavy-tailed columnar disorder. The findings have potential implications for understanding connectivity in heterogeneous materials and networks where rare but strong inhomogeneities govern global connectivity.

Abstract

We explore a bond percolation model on slabs $\mathbb{S}^+_k=\mathbb{Z}_+\times \mathbb{Z}_+\times\{0,\dots,k\}$ featuring one-dimensional inhomogeneities. In this context, a vertical column on the slab comprises the set of vertical edges projecting to the same vertex on $\mathbb{Z}_+\times\{0,\dots,k\}$. Columns are chosen based on the arrivals of a renewal process, where the tail distributions of inter-arrival times follow a power law with exponent $φ>1$. Inhomogeneities are introduced as follows: vertical edges on selected columns are open (closed) with probability $q$ (respectively $1-q$), independently. Conversely, vertical edges within unselected columns and all horizontal edges are open (closed) with probability $p$ (respectively $1-p$). We prove that for all sufficiently large $φ$ (depending solely on $k$), the following assertion holds: if $q>p_c(\mathbb{S}^+_k)$, then $p$ can be taken strictly smaller than $p_c(\mathbb{S}^+_k)$ in a manner that percolation still occurs.

Critical percolation on slabs with random columnar disorder

TL;DR

This work analyzes bond percolation on a 3D slab with columnar disorder driven by a heavy-tailed renewal process. The authors develop a multiscale renormalization tailored to heavy-tailed environments, proving that when the enhanced-column probability is supercritical and the renewal tail exponent is large enough (depending on the slab thickness ), percolation occurs at a horizontal bond probability strictly below the critical threshold . A key contribution is a polynomial upper bound on the slab correlation length in terms of , achieved via a combination of pivotal-edge analysis, Gluing Lemmas, and a 1-dependent renormalized percolation coupled to a Marcos-style multiscale scheme. The results illuminate near-critical behavior in disordered 3D systems with infinite-range vertical dependencies and provide a framework for extending near-critical percolation methods to slab geometries with heavy-tailed columnar disorder. The findings have potential implications for understanding connectivity in heterogeneous materials and networks where rare but strong inhomogeneities govern global connectivity.

Abstract

We explore a bond percolation model on slabs featuring one-dimensional inhomogeneities. In this context, a vertical column on the slab comprises the set of vertical edges projecting to the same vertex on . Columns are chosen based on the arrivals of a renewal process, where the tail distributions of inter-arrival times follow a power law with exponent . Inhomogeneities are introduced as follows: vertical edges on selected columns are open (closed) with probability (respectively ), independently. Conversely, vertical edges within unselected columns and all horizontal edges are open (closed) with probability (respectively ). We prove that for all sufficiently large (depending solely on ), the following assertion holds: if , then can be taken strictly smaller than in a manner that percolation still occurs.
Paper Structure (11 sections, 15 theorems, 116 equations, 10 figures)

This paper contains 11 sections, 15 theorems, 116 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Model and main results
  4. Notation
  5. Proof overview and tools
  6. An upper bound for the correlation length on slabs
  7. Proof of Proposition \ref{['cotaPivotal']}
  8. Proof of Proposition \ref{['theoremEdge']}
  9. Multiscale scheme
  10. Proof of Proposition \ref{['TeoDominacao']}
  11. The multiscale

Key Result

Theorem 1

Consider the percolation model $\mathbb{P}_{p,q}^\Lambda$ on the slab $\mathbb{S}^+_k$, with $\Lambda$ given by an $\phi$-renewal process with law $\nu_{\phi}$. There exists $\phi_0 = \phi_0(k)$ such that, for every $\phi>\phi_0$ and every $\phi$-renewal process, the following holds: for all $\varep

Figures (10)

  • Figure 1: Sketch of the sets $B_n,LB_n,RB_n, LS_n,RS_n,BS_n$ and $TS_n$.
  • Figure 2: Two-dimensional sketch of $D_n(0)$ (continuous line), $H_n(0)$ (dotted dark line) and $V_n(0)$ (dashed light line).
  • Figure 3: A sketch of the sets $\mathbf{B}_n$ and $C(f)$.
  • Figure 4: The renormalized percolation and the original equivalent.
  • Figure 5: The boxes $\mathbf{B}_N$, $R'_N$, $R_N"$ and $Q_N$.
  • ...and 5 more figures

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Definition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • proof : Proof of Theorem \ref{['corr_length_2']}
  • ...and 21 more