Abstract: 
We consider higherdimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam model Xk(n, p) of random kdimensional simplicial complexes on n vertices. We show that for p = Ω(logn/n), the eigenvalues of each of the matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of (k  2)dimensional faces. Garland’s result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of kdimensional simplices. In the second part of the paper, we apply this to the question of possible higherdimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the eigenvalues of the higherdimensional Laplacian capture the notion of coboundary expansion—a higherdimensional generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov; this question was raised, for instance, by Dotterrer and Kahle. We show that this most straightforward version of a higherdimensional discrete Cheeger inequality fails, in quite a strong way: For every k ≥ 2 and n ∈ N, there is a kdimensional complex Yn k on n vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised kdimensional Laplacian lie in the interval [1−O(1/√1), 1+0(1/√1]) but whose coboundary expansion is bounded from above by O(log n/n) and so tends to zero as n → ∞; moreover, Yn k can be taken to have vanishing integer homology in dimension less than k.
