Sparse random matrices: the eigenvalue spectrum revisited
Abstract
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.
Cite
@article{arxiv.cond-mat/0202406,
title = {Sparse random matrices: the eigenvalue spectrum revisited},
author = {Guilhem Semerjian and Leticia F. Cugliandolo},
journal= {arXiv preprint arXiv:cond-mat/0202406},
year = {2009}
}
Comments
18 pages, 7 figures. Accepted version, minor corrections, references added