Abstract:
In this talk, we introduce the spherical ensemble and more generally, the ratio of two independent random matrices with i.i.d. entries. We begin by reviewing some previous results for a single random matrix with i.i.d. entries. We then show that, for any test function with finitely many logarithmic singularities, the linear statistics of the spherical ensemble converge to a Gaussian distribution, after a suitable normalization by 1/\sqrt{log n}. The limiting distribution depends only on the weights of singularities. As an application, we obtain the finite-dimensional Gaussian convergence of the logarithm of the characteristic polynomial, normalization by 1/\sqrt{log n}. Moreover, we explicitly compute the variance and covariance without the 1/\sqrt{log n} normalization, showing that the field is log-correlated. All these results extend beyond spherical ensemble under a four moment matching condition. This is based on joint work with Djalil Chafaï and David García-Zelada.
Biography:
Yuanyuan Xu is a tenure-track assistant professor at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. She received her Ph.D. in applied mathematics from the University of California, Davis, and subsequently held postdoctoral positions at KTH Royal Institute of Technology and the Institute of Science and Technology Austria. Her research interests lie in probability and random matrix theory.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
This event is open to the NYU Shanghai community and Math community.