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

Philippe Sosoe (Princeton University)
Percy Wong (D.E. Shaw & Co.)


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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Pages: 1-18

Publication Date: March 15, 2014

DOI: 10.1214/EJP.v19-2638


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