# Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble

## TL;DR

This paper establishes an optimal lower bound for the least singular value of large shifted Ginibre matrices, especially near the spectral edge, improving previous bounds and employing advanced superbosonization techniques.

## Contribution

It provides the first optimal lower tail estimate for the least singular value of shifted Ginibre matrices at the spectral edge, using superbosonization.

## Key findings

- Optimal lower tail estimate derived
- Improved bounds over classical results
- Applicable to real and complex Gaussian 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 Br\'ezin-Hikami formulas in the real case, we rely on the superbosonization formula [Littelmann, Sommers, Zirnbauer, 2008].