Abstract: 
Eigenvalues associated to graphs are a wellstudied subject. In particular the spectra of the adjacency matrix and of the Laplacian of random graphs G(n, p) are known quite precisely. We consider generalizations of these matrices to simplicial complexes of higher dimensions and study their eigenvalues for the LinialMeshulam model X k(n, p) of random kdimensional simplicial complexes on n vertices. We show that for p = Ω(log n/n), the eigenvalues of both, the higherdimensional adjacency matrix and the Laplacian, are a.a.s. sharply concentrated around two values. In a second part of the paper, we discuss a possible higherdimensional analogue of the Discrete Cheeger Inequality. This fundamental inequality expresses a close relationship between the eigenvalues of a graph and its combinatorial expansion properties; in particular, spectral expansion (a large eigenvalue gap) implies edge expansion. Recently, a higherdimensional analogue of edge expansion for simplicial complexes was introduced by Gromov, and independently by Linial, Meshulam and Wallach and by Newman and Rabinovich. It is natural to ask whether there is a higherdimensional version of Cheeger's inequality. We show that the most straightforward version of a higherdimensional Cheeger inequality fails: for every k > 1, there is an infinite family of kdimensional complexes that are spectrally expanding (there is a large eigenvalue gap for the Laplacian) but not combinatorially expanding.
