Pairwise Negative Correlation for Uniform Spanning Subgraphs of the Complete Graph

TL;DR

This paper proves that certain uniform distributions over spanning subgraphs of complete graphs exhibit pairwise negative correlation when the number of vertices is large, extending previous results and supporting conjectures in negative dependence.

Contribution

It establishes the p-NC property for uniform measures on connected subgraphs, forests, and truncated subgraphs of complete graphs, for sufficiently large n.

Findings

p-NC holds for connected spanning subgraphs when n is large

p-NC verified for forests with fixed components

First proof of p-NC for connected subgraphs on complete graphs

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.