Benjamini-Schramm convergence and limiting eigenvalue density of random matrices
Abstract
We review the application of the notion of local convergence on locally finite randomly rooted graphs, known as Benjamini-Schramm convergence, to the calculation of the global eigenvalue density of random matrices from the beta-Gaussian and beta-Laguerre ensembles. By regarding a random matrix as the weighted adjacency matrix of a graph, and choosing the root of such a graph with uniform probability, one can use the Benjamini-Schramm limit to produce the spectral measure of the adjacency operator of the limiting graph. We illustrate how the Wigner semicircle law and the Marchenko-Pastur law are obtained from this machinery.
Cite
@article{arxiv.1702.01281,
title = {Benjamini-Schramm convergence and limiting eigenvalue density of random matrices},
author = {Sergio Andraus},
journal= {arXiv preprint arXiv:1702.01281},
year = {2018}
}
Comments
8 pages, 2 figures. Proceedings paper for the Probability Theory Symposium 2016 held at RIMS, Kyoto University, on Dec. 19-22 2016