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Percolation of words on the hypercubic lattice with one-dimensional long-range interactions

Pablo A. Gomes, Otávio Lima, Roger W C Silva

TL;DR

The paper addresses word percolation on a hypercubic lattice with one-dimensional long-range bonds. By coupling the long-range model to a highly supercritical oriented percolation process in the plane and introducing a finite truncation level $K$, the authors prove that if the long-range connection probabilities satisfy $\sum_n p_n=\infty$, then for any $p\in(0,1)$, $\varepsilon>0$, and $\alpha>0$, there exists $K$ with $\mathbb{P}_{p,\varepsilon}^K(W_0=\Xi)>1-\alpha$, implying all words can be seen from some vertex with probability 1. The core method combines a dynamical exploration with a block-wise coupling to an independent oriented percolation process, controlled by auxiliary estimates (Proposition) that ensure propagation dominates a supercritical regime. This work extends prior results on word percolation by showing that infinitude of long-range connections is not essential for almost-sure visibility of all words when a finite truncation is allowed, highlighting robust connectivity phenomena in long-range percolation settings.

Abstract

We investigate the problem of percolation of words in a random environment. To each vertex, we independently assign a letter $0$ or $1$ according to Bernoulli r.v.'s with parameter $p$. The environment is the resulting graph obtained from an independent long-range bond percolation configuration on $\mathbb{Z}^{d-1} \times \mathbb{Z}$, $d\geq 3$, where each edge parallel to $\mathbb{Z}^{d-1}$ has length one and is open with probability $ε$, while edges of length $n$ parallel to $\mathbb{Z}$ are open with probability $p_n$. We prove that if the sum of $p_n$ diverges, then for any $ε$ and $p$, there is a $K$ such that all words are seen from the origin with probability close to $1$, even if all connections with length larger than $K$ are suppressed.

Percolation of words on the hypercubic lattice with one-dimensional long-range interactions

TL;DR

The paper addresses word percolation on a hypercubic lattice with one-dimensional long-range bonds. By coupling the long-range model to a highly supercritical oriented percolation process in the plane and introducing a finite truncation level , the authors prove that if the long-range connection probabilities satisfy , then for any , , and , there exists with , implying all words can be seen from some vertex with probability 1. The core method combines a dynamical exploration with a block-wise coupling to an independent oriented percolation process, controlled by auxiliary estimates (Proposition) that ensure propagation dominates a supercritical regime. This work extends prior results on word percolation by showing that infinitude of long-range connections is not essential for almost-sure visibility of all words when a finite truncation is allowed, highlighting robust connectivity phenomena in long-range percolation settings.

Abstract

We investigate the problem of percolation of words in a random environment. To each vertex, we independently assign a letter or according to Bernoulli r.v.'s with parameter . The environment is the resulting graph obtained from an independent long-range bond percolation configuration on , , where each edge parallel to has length one and is open with probability , while edges of length parallel to are open with probability . We prove that if the sum of diverges, then for any and , there is a such that all words are seen from the origin with probability close to , even if all connections with length larger than are suppressed.
Paper Structure (10 sections, 5 theorems, 51 equations, 2 figures)

This paper contains 10 sections, 5 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The long-range setting
  3. Main result
  4. Statement
  5. Tools and proof of Theorem \ref{['words']}
  6. The coupling
  7. Auxiliary result
  8. Proof of Proposition \ref{['aux']}
  9. Proofs of Lemmas
  10. Acknowledgements

Key Result

Theorem 1

Consider a long-range oriented percolation process on $\mathbb{G}^d_+$, $d\geqslant 3$, and assume $\{p_n\}_{n\in{\mathbb{N}}}$ satisfies diverge. Then, for all $p\in(0,1)$, $\epsilon>0$, and $\alpha>0$, there exists $K=K(\{p_n\},p,\epsilon,\alpha)\in{\mathbb{N}}$ large enough such that

Figures (2)

  • Figure 1: Graphical representation of the sets $L_{16m,1}$, $L_{16m,2}$, and $L_{16m,3}$.
  • Figure 2: Graphical representation of the event $E_1\cap E_2\cap E_3$.

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Theorem 2: Ligget T.M. and Steif J.F., 2006