Convergence of the eigenvalue density for beta-Laguerre ensembles on short scales
Probability
2014-04-08 v3
Abstract
In this note, we prove that the normalized trace of the resolvent of the beta-Laguerre ensemble eigenvalues is close to the Stieltjes transform of the Marchenko-Pastur (MP) distribution with very high probability, for values of the imaginary part greater than m^{-1+\epsilon}. As an immediate corollary, we obtain convergence of the one-point density to the MP law on short scales. The proof serves to illustrate some simplifications of the method introduced in our previous work to prove a local semi-circle law for Gaussian beta-ensembles.
Keywords
Cite
@article{arxiv.1302.1458,
title = {Convergence of the eigenvalue density for beta-Laguerre ensembles on short scales},
author = {Philippe Sosoe and Percy Wong},
journal= {arXiv preprint arXiv:1302.1458},
year = {2014}
}
Comments
Various corrections based on referee comments. To appear in Electron. J. Probab