Matchings on trees and the adjacency matrix: A determinantal viewpoint.

Let G$$ G $$ be a finite tree. For any matching M$$ M $$ of G$$ G $$, let U(M)$$ U(M) $$ be the set of vertices uncovered by M$$ M $$. Let ℳG$$ {\mathcal{M}}_G $$ be a uniform random maximum size matching of G$$ G $$. In this paper, we analyze the structure of U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$. We first show that U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$ is a determinantal process. We also show that for most vertices of G$$ G $$, the process U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$ 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(ℳG)$$ U\left({\mathcal{M}}_G\right) $$ can be also well approximated using the local structure of G$$ G $$. In other words, in the realm of trees, the normalized Shannon entropy of U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$—that is, the normalized logarithm of the number of maximum size matchings of G$$ G $$—is a Benjamini‐Schramm continuous parameter. [ABSTRACT FROM AUTHOR]