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2014 Convergence of the eigenvalue density for Laguerre beta ensembles on short scales
Philippe Sosoe, Percy Wong
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Electron. J. Probab. 19: 1-18 (2014). DOI: 10.1214/EJP.v19-2638

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 $m^{1+\varepsilon}$. 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.

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Philippe Sosoe. Percy Wong. "Convergence of the eigenvalue density for Laguerre beta ensembles on short scales." Electron. J. Probab. 19 1 - 18, 2014. https://doi.org/10.1214/EJP.v19-2638

Information

Accepted: 15 March 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1310.60004
MathSciNet: MR3183578
Digital Object Identifier: 10.1214/EJP.v19-2638

Subjects:
Primary: 60B20

Keywords: Beta ensembles , Marchenko-Pastur law , Ranbom Matrices

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