论文标题
随机施纳仪中的简单跨越树
Simplicial spanning trees in random Steiner complexes
论文作者
论文摘要
图$ g $中的生成树$ t $是$ g $的子图，其顶点集与$ g $，这是一棵树。 1981年，麦凯（McKay）证明了在随机$ k $ groumard图中跨越树木数量的渐近结果。在本文中，我们证明了McKay对随机$ d $维的结果的高度概括，$ k $ - $ n $ dertices上的规范简单复合物表明，简化的简单跨越的树是$（ξ_{ξ_{dem {d，k}+n} $ ntty^$ nty} $ { $ξ_{d，k} $是一个明确的常数，提供$ k> 4d^2+d+2 $。我们证明的一个关键要素是，这种随机复合物与$ d $ d $ $ k $ to-k $的植木综合体的局部收敛，这使我们能够将麦凯的结果推广到Kesten-McKay分布中。
A spanning tree $T$ in a graph $G$ is a sub-graph of $G$ with the same vertex set as $G$ which is a tree. In 1981, McKay proved an asymptotic result regarding the number of spanning trees in random $k$-regular graphs. In this paper we prove a high-dimensional generalization of McKay's result for random $d$-dimensional, $k$-regular simplicial complexes on $n$ vertices, showing that the weighted number of simplicial spanning trees is of order $(ξ_{d,k}+o(1))^{\binom{n}{d}}$ as $n\to\infty$, where $ξ_{d,k}$ is an explicit constant, provided $k> 4d^2+d+2$. A key ingredient in our proof is the local convergence of such random complexes to the $d$-dimensional, $k$-regular arboreal complex, which allows us to generalize McKay's result regarding the Kesten-McKay distribution.
