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. 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 -faces. Furthermore, we consider a random walk on such a complex, which generalizes the standard random walk on a graph. We show that the associated conductance is with high probability bounded away from 0, resulting in a bound on the mixing time that is logarithmic in the number of vertices of the complex.

中文翻译：

---

随机单纯复形中的代数和组合展开

在本文中，我们考虑了在上同调连通性阈值之上的d维 Linial-Meshulam 随机单纯复形的组合拉普拉斯算子的扩展性质和谱。我们考虑由 Parzanchevski、Rosenthal 和 Tessler 引入的 Laplace 算子的谱间隙和 Cheeger 常数。我们表明，随机单纯复形的谱间隙以及 Cheeger 常数都集中在所有-面孔。此外，我们考虑在这样一个复杂的随机游走，它概括了图上的标准随机游走。我们表明，相关的电导很有可能远离 0，从而导致混合时间的界限与复合体的顶点数成对数。