Eigenvalues Outside the Bulk of Inhomogeneous Erdős–Rényi Random Graphs

Article Type

Research Article

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Journal of Statistical Physics


In this article, an inhomogeneous Erdős–Rényi random graph on { 1 , … , N} is considered, where an edge is placed between vertices i and j with probability εNf(i/ N, j/ N) , for i≤ j, the choice being made independently for each pair. The integral operator If associated with the bounded function f is assumed to be symmetric, non-negative definite, and of finite rank k. We study the edge of the spectrum of the adjacency matrix of such an inhomogeneous Erdős–Rényi random graph under the assumption that NεN→ ∞ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by NεN, is compactly supported, the kth largest eigenvalue goes to infinity. It turns out that the largest eigenvalue after appropriate scaling and centering converges to a Gaussian law, if the largest eigenvalue of If has multiplicity 1. If If has k distinct non-zero eigenvalues, then the joint distribution of the k largest eigenvalues converge jointly to a multivariate Gaussian law. The first order behaviour of the eigenvectors is derived as a byproduct of the above results. The results complement the homogeneous case derived by [18].

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