ON-LINE and on-campus QLunch: Sungsoo Byun

Speaker: Sungsoo Byun, Korea Institute for Advanced Study (KIAS).

Title: Real eigenvalues of elliptic random matrices

Abstract: In this talk, I will discuss the real eigenvalues of the real elliptic Ginibre matrix, the model which provides a natural bridge between Hermitian and non-Hermitian random matrix theories. In the maximally non-Hermitian regime, which corresponds to the matrix model with real i.i.d. Gaussian entries, it was pioneered by Edelman, Kostlan, and Shub that the number of real eigenvalues is of order \sqrt{N}, where N is the size of the matrix. Moreover, it can be heuristically conjectured that as a real random matrix becomes more symmetric, it gets more real eigenvalues.

I will demonstrate that such a statement can be made rigorous by presenting the large-N expansion of the mean and the variance of the number of real eigenvalues in the almost-Hermitian regime, where one can observe a non-trivial transition between real i.i.d. and real symmetric random matrices. Furthermore I will explain the limiting empirical distributions of the real eigenvalues which interpolate the Wigner semicircle law and the uniform distribution. The proofs are based on the skew-orthogonal representation of the correlation kernel due to Forrester and Nagao.

This is a joint work with Nam-Gyu Kang (KIAS), Ji Oon Lee (KAIST) and Jinyeop Lee (LMU).