Matchings on trees and the adjacency matrix: A determinantal viewpoint

TL;DR

This paper investigates the structure of uncovered vertices in maximum matchings on trees, revealing a determinantal process linked to the adjacency matrix and showing local approximation and entropy properties.

Contribution

It establishes that the uncovered vertices form a determinantal process and connects this to the adjacency matrix, providing new insights into maximum matchings on trees.

Findings

U(M_G) is a determinantal process.

Local neighborhoods can approximate U(M_G).

Normalized Shannon entropy is a continuous parameter.

Abstract

Let $G$ be a finite tree. For any matching $M$ of $G$, let $U(M)$ be the set of vertices uncovered by $M$. Let $\mathcal{M}_G$ be a uniform random maximum size matching of $G$. In this paper, we analyze the structure of $U(\mathcal{M}_G)$. We first show that $U(\mathcal{M}_G)$ is a determinantal process. We also show that for most vertices of $G$, the process $U(\mathcal{M}_G)$ in a small neighborhood of that vertex can be well approximated based on a somewhat larger neighborhood of the same vertex. Then we show that the normalized Shannon entropy of $U(\mathcal{M}_G)$ can be also well approximated using the local structure of $G$. In other words, in the realm of trees, the normalized Shannon entropy of $U(\mathcal{M}_G)$ -- that is, the normalized logarithm of the number of maximum size matchings of $G$ -- is a Benjamini-Schramm continuous parameter. We show that $U(\mathcal{M}_G)$ is a determinantal process through establishing a new connection between $U(\mathcal{M}_G)$ and the adjacency matrix of $G$. This result sheds a new light on the well-known fact that on a tree, the number of vertices uncovered by a maximum size matching is equal to the nullity of the adjacency matrix. Some of the proofs are based on the well established method of introducing a new perturbative parameter, which we call temperature, and then define the positive temperature analogue of $\mathcal{M}_G$, the so called monomer-dimer model, and let the temperature go to zero.