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Critical threshold for regular graphs

Ishaan Bhadoo

TL;DR

The paper investigates the sharp percolation threshold $p_c$ for $d$-regular graphs, establishing that $p_c(G) \ge \frac{1}{d-1}$ with equality on the $d$-regular tree, and proves that among quasi-transitive $d$-regular graphs this equality holds if and only if the graph is a tree. The authors employ a covering-graph framework, showing that a strong covering map from the $d$-regular tree to any quasi-transitive graph with cycles yields a strict inequality $p_c(G) > \frac{1}{d-1}$ unless $G$ is a tree, by leveraging results on coverings due to Martineau and Severo. They also exhibit non-quasi-transitive counterexamples demonstrating that the quasi-transitive hypothesis is essential. The connective-constant perspective and branching-number theory anchor the analysis, linking growth properties to critical thresholds and enabling tractable treatment of trees and subperiodic structures.

Abstract

In this article, we study the critical percolation threshold $p_c$ for $d$-regular graphs. It is well-known that $p_c \geq \frac{1}{d-1}$ for such graphs, with equality holding for the $d$-regular tree. We prove that among all quasi-transitive $d$-regular graphs, the equality $p_c(G) = \frac{1}{d-1}$ holds if and only if $G$ is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption.

Critical threshold for regular graphs

TL;DR

The paper investigates the sharp percolation threshold for -regular graphs, establishing that with equality on the -regular tree, and proves that among quasi-transitive -regular graphs this equality holds if and only if the graph is a tree. The authors employ a covering-graph framework, showing that a strong covering map from the -regular tree to any quasi-transitive graph with cycles yields a strict inequality unless is a tree, by leveraging results on coverings due to Martineau and Severo. They also exhibit non-quasi-transitive counterexamples demonstrating that the quasi-transitive hypothesis is essential. The connective-constant perspective and branching-number theory anchor the analysis, linking growth properties to critical thresholds and enabling tractable treatment of trees and subperiodic structures.

Abstract

In this article, we study the critical percolation threshold for -regular graphs. It is well-known that for such graphs, with equality holding for the -regular tree. We prove that among all quasi-transitive -regular graphs, the equality holds if and only if is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption.

Paper Structure

This paper contains 10 sections, 7 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Connection to the connective constant
  3. Percolation on trees
  4. Branching number and the critical point for trees
  5. Superiodic trees
  6. Non quasi-transitive counter-examples
  7. Proof of the Theorem
  8. Critical points under coverings
  9. Concluding remarks
  10. Acknowledgements

Key Result

Theorem 2

Let $G$ be a quasi-transitive $d$-regular graph. Then $p_c(G) \geq \frac{1}{d - 1}$ and equality holds if and only if $G$ is a tree.

Figures (1)

  • Figure 1: On removing the edge $e$ we get a sub-periodic tree $T$ with gr$T=$ br$T=d -1$

Theorems & Definitions (10)

  • Definition 1
  • Theorem 2
  • Lemma 3
  • proof
  • Theorem 4: G. Grimmett, Z. Li, grimmett2015bounds
  • Theorem 5: R. Lyons, lyons1990random
  • Theorem 6: Subperiodicity and Branching Number, lyons2017probability
  • Theorem 7: F.Severo, S. Martineau, martineau2019strict
  • Proposition 8
  • proof