Eigenvalue distribution of large dilute random matrices

Abstract

We study the eigenvalue distribution of dilute N3N random matrices HN that in the pure ~undiluted! case describe the Hopfield model. We prove that for the fixed dilution parameter a the normalized counting function ~NCF! of HN converges as N!` to a unique sa(l). We find the moments of this distribution explicitly, analyze the 1/a correction, and study the asymptotic properties of sa(l) for large ulu. We prove that sa(l) converges as a !` to the Wigner semicircle distribution ~SCD!. We show that the SCD is the limit of the NCF of other ensembles of dilute random matrices. This could be regarded as evidence of stability of the SCD to dilution, or more generally, to random modulations of large random matrices.