Precise asymptotics for the spectral radius of a large random matrix

TL;DR

This paper derives precise three-term asymptotics for the spectral radius of large random matrices with i.i.d. entries, surpassing the circular law, and introduces a new decorrelation technique for singular values.

Contribution

It provides the first detailed asymptotic expansion for the spectral radius of large random matrices, using novel decorrelation methods for singular values.

Findings

Spectral radius asymptotics with three-term precision

Universal coefficients differing from previous eigenvalue asymptotics

New decorrelation mechanism for low-lying singular values

Abstract

We consider the spectral radius of a large random matrix $X$ with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of $X$ in [29]. To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of $X-z$ for different complex shift parameters $z$ using the Dyson Brownian Motion.