Single eigenvalue fluctuations of random matrices

VIENNA PROBABILITY SEMINAR

Gustavsson proved that the fluctuations of a bulk GUE eigenvalue around its mean are asymptotically Gaussian after a suitable rescaling.  O'Rourke extended this to the GOE and GSE using a coupling of Forrester and Rains.  In this talk we present recent results on the universality of these fluctuations for other classes of random matrices, including matrices of general Wigner-type under a one-cut assumption.  We use as input the homogenization theory of Dyson Brownian motion of L.-Sosoe-Yau as well as the works on the theory of the quadratic vector equation and general Wigner-type matrices of Ajanki-Erd\H{o}s-Kr{\"u}ger.  Based on joint work with P. Lopatto and P. Sosoe.