Modul: MAT971 Stochastische Prozesse

## Hermitian Cusp Universality and Non-Hermitian Edge Universality

Dr. Dominik Schröder talk

**Date:** 02.10.19 **Time:** 17.15 - 18.15 **Room:** ETH HG G 43

In the last decade, the Wigner-Dyson-Mehta (WDM) universality conjecture has been proven for very general Hermitian random matrix ensembles in the bulk and at the edge of the self consistent density of states (scDOS). The scDOS of Wigner-type matrices can, in addition, also feature cubic root cusp singularities for which we prove universality in the real and complex symmetry class through an optimal local law and the fast relaxation to equilibrium of Dyson Brownian motion. This result completes the last remaining case of the WDM conjecture for Wigner-type random matrices. About universality for non-Hermitian matrices much less is known. We prove that for non-Hermitian i.i.d. matrices the local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements are Gaussian. The proof relies on the optimal local law in the cusp regime of Hermitian random matrices, and a supersymmetric estimate on the least singular value of shifted Ginibre ensembles. This estimate on the least singular value improves the classical smoothing bound from [Sankar, Spielman, Teng (2006)] in the transitional edge regime.