Convergence of the eigenvalue density for Laguerre beta ensembles on short scales. (English) Zbl 1310.60004

Sosoe, Philippe; Wong, Percy

Electron. J. Probab. 19, Paper No. 34, 18 p. (2014).

Summary: 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.

MSC:

60B20

Random matrices (probabilistic aspects)

Keywords:

random matrices; beta ensembles; Marchenko-Pastur law