论文标题
随机简单复合物中的代数和组合扩展
Algebraic and combinatorial expansion in random simplicial complexes
论文作者
论文摘要
在本文中，我们考虑了$ d $二维的外线式汇总随机简单复合物的膨胀属性和组合拉普拉斯操作员的频谱，在同时理性连接阈值之上。我们考虑了Laplace操作员和Cheeger Constant的光谱差距，因为Parzanchevski，Rosenthal和Tessler引入了这一点（$ Combinatorica $ 36，2016年）。我们表明，具有很高的概率，随机简单络合物以及Cheeger常数的光谱间隙都集中在所有$ d-1 $ faces中的最低共同度周围。此外，我们考虑对如此复杂的随机行走进行概括，并表明相关的电导范围具有很高的概率，远离0。
In this paper we consider the expansion properties and the spectrum of the combinatorial Laplace operator of a $d$-dimensional Linial-Meshulam random simplicial complex, above the cohomological connectivity threshold. We consider the spectral gap of the Laplace operator and the Cheeger constant as this was introduced by Parzanchevski, Rosenthal and Tessler ($Combinatorica$ 36, 2016). We show that with high probability the spectral gap of the random simplicial complex as well as the Cheeger constant are both concentrated around the minimum co-degree of among all $d-1$-faces. Furthermore, we consider a generalisation of a random walk on such a complex and show that the associated conductance is with high probability bounded away from 0.
