Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative Supercritical Bounds for Disconnection in Bernoulli Site Percolation

Zhongyang Li

TL;DR

The paper addresses the problem of obtaining quantitative disconnection tails in supercritical Bernoulli site percolation on general infinite graphs. It introduces the $(p,\varepsilon,c)$-packing number $\mathrm{PK}_{p,\varepsilon,c}(S)$ together with the local functional $\varphi_p^v(S)$ to connect finite-volume data to the global percolation threshold, and proves an explicit exponential-type bound on $\mathbb{P}_{p}(S\nleftrightarrow\infty)$ that scales with the packing size. The method combines a finite-volume connectivity analysis (FCA) with an independence amplification across well-separated packing vertices, yielding a bound of the form $\mathbb{P}_p(S\nleftrightarrow\infty) \le \frac{(1-c)\varepsilon}{c}+(1+\varepsilon)(1-c)^{\mathrm{PK}_{p,\varepsilon,c}(S)}$ and its infimum representation. By situating the result within existing effective supercritical bounds (capacity-based, isoperimetric, and conjectural isoperimetric tails), the work provides a unified, quantitative framework for disconnection tails driven by packing structure.

Abstract

For any infinite, connected, locally finite graph $G=(V,E)$, any parameter $p>p^{\mathrm{site}}_{c}(G)$, and any (finite or infinite) set of vertices $S\subset V$, we derive explicit exponential-type upper bounds on the disconnection probability $\mathbb{P}_{p}(S\nleftrightarrow\infty)$. The estimates are expressed in terms of a packing profile of $S$, encoded by a $(p,\varepsilon,c)$--packing number, which counts how many well-separated vertices in $S$ exhibit controlled local-to-global connectivity. The proof combines a local functional characterization of $p^{\mathrm{site}}_{c}$ from \cite{ZL24,ZL26} with a packing construction and an amplification-by-independence argument, in the direction of Problem~1.6 in \cite{DC20}.

Quantitative Supercritical Bounds for Disconnection in Bernoulli Site Percolation

TL;DR

The paper addresses the problem of obtaining quantitative disconnection tails in supercritical Bernoulli site percolation on general infinite graphs. It introduces the -packing number together with the local functional to connect finite-volume data to the global percolation threshold, and proves an explicit exponential-type bound on that scales with the packing size. The method combines a finite-volume connectivity analysis (FCA) with an independence amplification across well-separated packing vertices, yielding a bound of the form and its infimum representation. By situating the result within existing effective supercritical bounds (capacity-based, isoperimetric, and conjectural isoperimetric tails), the work provides a unified, quantitative framework for disconnection tails driven by packing structure.

Abstract

For any infinite, connected, locally finite graph , any parameter , and any (finite or infinite) set of vertices , we derive explicit exponential-type upper bounds on the disconnection probability . The estimates are expressed in terms of a packing profile of , encoded by a --packing number, which counts how many well-separated vertices in exhibit controlled local-to-global connectivity. The proof combines a local functional characterization of from \cite{ZL24,ZL26} with a packing construction and an amplification-by-independence argument, in the direction of Problem~1.6 in \cite{DC20}.
Paper Structure (13 sections, 3 theorems, 27 equations)

This paper contains 13 sections, 3 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Model and the quantitative supercritical problem
  3. Relation to effective supercritical estimates
  4. (i) Problem 1.6 in DC20.
  5. (ii) Item 8 in EST25.
  6. (iii) Conjecture 5.1 in HH21.
  7. Main result: an exponential bound via a packing profile
  8. Method: FCA + functional characterization of $p^{\mathrm{site}}_{c}$
  9. Organization
  10. A characterization of critical percolation probability $p_c^{\mathrm{site}}$
  11. Definition of the $\varphi_p^v(S)$-functional
  12. Characterization theorem from ZL24
  13. Packing definition and quantitative upper bounds

Key Result

Theorem 1.1

Let $G=(V,E)$ be an infinite, connected, locally finite graph, let $p>p^{\mathrm{site}}_{c}(G)$, and let $S\subset V$ be any (finite or infinite) set of vertices. Then

Theorems & Definitions (6)

  • Theorem 1.1: Quantitative supercritical disconnection bound
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof