Electronic Journal of Probability

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

Philippe Sosoe and Percy Wong

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Ranbom Matrices Beta Ensembles Marchenko-Pastur law

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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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