# The local relaxation flow approach to universality of the local statistics for random matrices

 Title: The local relaxation flow approach to universality of the local statistics for random matrices Author: Schlein, Benjamin; Yin, Jun; Yau, Horng-Tzer Note: Order does not necessarily reflect citation order of authors. Citation: Erdos, László, Benjamin Schlein, Horng-Tzer Yau, and Jun Yin. 2012. “The Local Relaxation Flow Approach to Universality of the Local Statistics for Random Matrices.” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 48 (1) (February): 1–46. doi:10.1214/10-aihp388. Full Text & Related Files: 0911.3687v5.pdf (569.2Kb; PDF) Abstract: We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues {$$x_{j}$$}$$_{j=1}^{N}$$ are close to their classical location {$$\gamma$$$$_{j}$$}$$_{j=1}^{N}$$ determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1/$$N$$, the necessary apriori estimate on the location of eigenvalues requires only to know that $$\mathbb{E}$$ |$$x_{j}$$ $$-$$ $$\gamma$$$$_{j}$$|$$^{2}$$ $$\leq$$ $$N$$$$^{-1-\epsilon}$$ on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for Wishart matrices. Published Version: doi://10.1214/10-AIHP388 Other Sources: http://arxiv.org/abs/0911.3687v5 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:25426537 Downloads of this work: