# Optimal lower bound on the least singular value of the shifted Ginibre ensemble

Giorgio Cipolloni, László Erdős, Dominik Schröder

Probab. Math. Phys.Vol. 1 (2020)

## Summary

Using the superbosonization technique we derive optimal lower bounds on the least singular value of shifted Ginibre random matrices.## Abstract

We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant $z\in\mathbb{C}$. We prove an optimal lower tail estimate on this singular value in the critical regime where $z$ is around the spectral edge thus improving the classical bound of [Sankar, Spielman, Teng, 2006] in the edge regime. Lacking Brézin-Hikami formulas in the real case, we rely on the superbosonization formula [Littelmann, Sommers, Zirnbauer, 2008].

## Paper

1908.01653.pdf