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

#### Electronic Journal of Probability

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 Field Value Title Convergence of the eigenvalue density for Laguerre beta ensembles on short scales Creator Sosoe, Philippe; Princeton University Wong, Percy; D.E. Shaw & Co. Subject Ranbom Matrices, Beta Ensembles, Marchenko-Pastur law 60B20 Description 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. Publisher Electronic Journal of Probability Contributor NSERC, NSF Date 2014-01-02 Type Peer-reviewed Article Format application/pdf Identifier http://ejp.ejpecp.org/article/view/2638 10.1214/EJP.v19-2638 Source Electronic Journal of Probability; Vol 19; 1-18 Language en Relation http://ejp.ejpecp.org/article/downloadSuppFile/2638/15012 http://ejp.ejpecp.org/article/downloadSuppFile/2638/15013 http://ejp.ejpecp.org/article/downloadSuppFile/2638/16629 Rights The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.Summary of the Creative Commons Attribution LicenseYou are free to copy, distribute, display, and perform the work to make derivative works to make commercial use of the workunder the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author.