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On the Continuity of Enhancement Percolation

Paul Duncan, Benjamin Schweinhart, David Sivakoff

TL;DR

The article addresses whether enhancement percolation on $\mathbb{Z}^d$ exhibits continuity with respect to finite truncations of the enhancement family. It develops planar techniques based on sharp thresholds, torus reductions, and crossing results to prove $\lim_{k\to\infty} p_k^e = p_c^e$ in $d=2$ for symmetric connected enhancements, and provides a partial higher-dimensional theory. In $d\ge 3$, rotund enhancement families yield a finite-stage percolation result: for any $p>q_c$ there exists an $N$ with $P_N$ percolating, established via coupling to Poisson disk processes and $k$-dependent domination. Overall, the work connects enhancement-locality with multiscale and planar percolation techniques to characterize when infinite enhancement behavior can be captured by finite truncations, with implications for understanding long-range and pattern-driven connectivity in stochastic networks.

Abstract

We study bond percolation in $\mathbb{Z}^d$ with an unbounded family of enhancements that enable additional bonds to act as open. A natural question is whether percolation occurs in this model if and only if percolation also occurs in the system with a finite subcollection of enhancements. We give an affirmative answer in dimension $d=2$ for symmetric families of connected enhancements, and in dimensions $d\ge 3$ we prove a partial result.

On the Continuity of Enhancement Percolation

TL;DR

The article addresses whether enhancement percolation on exhibits continuity with respect to finite truncations of the enhancement family. It develops planar techniques based on sharp thresholds, torus reductions, and crossing results to prove in for symmetric connected enhancements, and provides a partial higher-dimensional theory. In , rotund enhancement families yield a finite-stage percolation result: for any there exists an with percolating, established via coupling to Poisson disk processes and -dependent domination. Overall, the work connects enhancement-locality with multiscale and planar percolation techniques to characterize when infinite enhancement behavior can be captured by finite truncations, with implications for understanding long-range and pattern-driven connectivity in stochastic networks.

Abstract

We study bond percolation in with an unbounded family of enhancements that enable additional bonds to act as open. A natural question is whether percolation occurs in this model if and only if percolation also occurs in the system with a finite subcollection of enhancements. We give an affirmative answer in dimension for symmetric families of connected enhancements, and in dimensions we prove a partial result.

Paper Structure

This paper contains 7 sections, 14 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Continuity for Enhancements in the Plane
  4. Sharpness
  5. Continuity below $q_c$
  6. Continuity above $q_c$
  7. Continuity in Higher Dimensions

Key Result

Theorem 2

Let $\mathscr{E}=\left\{\left( T_\alpha,S_\alpha \right)\right\}_{\alpha\in I}$ be a set of enhancements for bond percolation on $\mathbb{Z}^2$ so that $S_{\alpha}$ is a connected subgraph of the nearest-neighbor graph $\mathbb{Z}^2$ for each $\alpha\in I.$ Then

Figures (1)

  • Figure 1: An illustration of enhancement percolation with a single enhancement. $T$ is shown in blue and $S\setminus T$ is shown in orange. For every subgraph of Bernoulli percolation (shown in black) congruent to the blue graph an appropriately transformed collection of (lighter) orange edges is added. Here, a horizontal crossing occurs in the enhanced percolation but not in the original percolation.

Theorems & Definitions (26)

  • Theorem 2
  • Theorem 4
  • Theorem 5: Friedgut, Kalai
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • Corollary 8
  • proof
  • Lemma 9
  • ...and 16 more