Convergence of the eigenvalue density for beta-Laguerre 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+ε}. 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.

Various corrections based on referee comments. To appear in Electron. J. Probab

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Cited by in corpus (1)

• Beta Laguerre ensembles in global regime