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Pairwise Negative Correlation for Uniform Spanning Subgraphs of the Complete Graph

Pengfei Tang, Zibo Zhang

Abstract

We investigate the pairwise negative correlation (p-NC) property for uniform probability measures on several families of spanning subgraphs of the complete graph $K_n$. Motivated by conjectured negative dependence properties of the random-cluster model with $q<1$, we focus on three natural families: the set of all connected spanning subgraphs, the set of forests with exactly $k$ components, and the set of connected spanning subgraphs with excess $k$, where $k$ is a fixed integer. We prove that for each of these families, the associated uniform measure satisfies the p-NC property provided $n$ is sufficiently large. Our results extend earlier work on uniform forests and provide the first verification of the p-NC property for uniform connected subgraphs and their truncations on complete graphs.

Pairwise Negative Correlation for Uniform Spanning Subgraphs of the Complete Graph

Abstract

We investigate the pairwise negative correlation (p-NC) property for uniform probability measures on several families of spanning subgraphs of the complete graph . Motivated by conjectured negative dependence properties of the random-cluster model with , we focus on three natural families: the set of all connected spanning subgraphs, the set of forests with exactly components, and the set of connected spanning subgraphs with excess , where is a fixed integer. We prove that for each of these families, the associated uniform measure satisfies the p-NC property provided is sufficiently large. Our results extend earlier work on uniform forests and provide the first verification of the p-NC property for uniform connected subgraphs and their truncations on complete graphs.
Paper Structure (31 sections, 48 theorems, 424 equations, 3 figures)

This paper contains 31 sections, 48 theorems, 424 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Organization of the paper and a list of notation
  5. Uniform connected subgraphs of complete graphs
  6. Sketch of the proof of Theorem \ref{['thm: p-NC for UC']}
  7. Estimates of the disconnection probabilities
  8. Proof of Theorem \ref{['thm: p-NC for UC']}
  9. Uniform $k$-component forests of complete graphs
  10. Two observations using first and second moment arguments
  11. The number of $k$-forests of complete graphs
  12. Review of the $k$-forest formula due to Liu and ChowLiu_Chow1981
  13. Some preliminary lemmas
  14. The number of $k$-forests containing a pair of adjacent edges
  15. Proof of \ref{['thm: p-NC for kUF']}
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\mu$ be a strictly positive probability measure on $\Omega\coloneq \{0,1\}^E$ satisfying eq: PLC. Then for all increasing functions $X,Y: \Omega\to{\mathbb R} \,.$

Figures (3)

  • Figure 1: (a) The graph $G=K_n/e$; (b) The graph $G=K_n/\{e,f\}$ for a pair of adjacent edges $e,f$
  • Figure 2: Three Cases
  • Figure 3: A graph for which the uniform $2$-forest measure fails the p-NC property (adapted from HSW2022).

Theorems & Definitions (114)

  • Theorem 1.1: FKG
  • Conjecture 1.2: Kahn2000Grimmett_Winkler2004
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Definition 2.2
  • ...and 104 more