Algebraic and combinatorial expansion in random simplicial complexes

In this paper we consider the expansion properties and spectrum of combinatorial Laplace operator a d-dimensional Linial–Meshulam random simplicial complex, above cohomological connectivity threshold. We spectral gap Cheeger constant as was introduced by Parzanchevski, Rosenthal, Tessler. show that with high probability complex well are both concentrated around minimum co-degree among all -faces. Furthermore, walk on such which generalizes standard graph. associated conductance is bounded away from 0, resulting in bound mixing time logarithmic number vertices complex.