For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that λ(G) ≤ h(G), where λ(G) is the second smallest eigenvalue of the Laplacian of a graph G and h(G) is the Cheeger constant measuring the edge expansion of G. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs).

Eigenvalues associated to graphs are a well-studied 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 Linial--Meshulam model Xk(n,p) of random k-dimensional simplicial complexes on n vertices. We show that for p=Ω(log n/n), the eigenvalues of both, the higher-dimensional adjacency matrix and the Laplacian, are a.a.s.~sharply concentrated around two values.