## Electronic Journal of Probability

### Convergence of the eigenvalue density for Laguerre beta ensembles on short scales

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 34, 18 pp.

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

https://projecteuclid.org/euclid.ejp/1465065676

Digital Object Identifier
doi:10.1214/EJP.v19-2638

Mathematical Reviews number (MathSciNet)
MR3183578

Zentralblatt MATH identifier
1310.60004

Rights

#### Citation

Sosoe, Philippe; Wong, Percy. Convergence of the eigenvalue density for Laguerre beta ensembles on short scales. Electron. J. Probab. 19 (2014), paper no. 34, 18 pp. doi:10.1214/EJP.v19-2638. https://projecteuclid.org/euclid.ejp/1465065676

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