Sparse random matrices: the eigenvalue spectrum revisited

Details Sparse random matrices: the eigenvalue spectrum revisited Sparse random matrices: the eigenvalue spectrum revisited

2002-02-22

arxiv.org/abs/cond-mat/0202406v2

J. Phys. A 35 (2002) 4837

Physics

Condensed Matter

18 pages, 7 figures. Accepted version, minor corrections, references added

Scientific paper

10.1088/0305-4470/35/23/303

We revisit the derivation of the density of states of sparse random matrices. We derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit. Using the iterative scheme introduced by Biroli and Monasson [J. Phys. A 32, L255 (1999)] we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration. The combination of the two methods yields a very simple simple geometric interpretation of the tails of the spectrum. We test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum.

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