  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.
