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

中文翻译：

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树和邻接矩阵的匹配：决定性的观点

让$G$是一棵有限树。对于任何匹配$\mathrm{中号}$的$G$， 让$U（\mathrm{中号}）$是未被覆盖的顶点集$\mathrm{中号}$。让${ℳ}_G$是均匀随机最大尺寸匹配$G$。在本文中，我们分析了结构$U（{ℳ}_G）$。我们首先证明$U（{ℳ}_G）$是一个决定性的过程。我们还表明，对于大多数顶点$G$，过程$U（{ℳ}_G）$在该顶点的一个小邻域中，可以基于同一顶点的稍大的邻域来很好地近似。然后我们证明归一化香农熵$U（{ℳ}_G）$也可以使用局部结构很好地近似$G$。换句话说，在树木领域，归一化香农熵为$U（{ℳ}_G）$—即最大尺寸匹配数的归一化对数$G$— 是 Benjamini-Schramm 连续参数。