On the rightmost eigenvalue of non-Hermitian random matrices

TL;DR

This paper derives a precise asymptotic expansion for the rightmost eigenvalue of large non-Hermitian random matrices with i.i.d. complex entries, including an optimal error estimate, demonstrating universality of all terms.

Contribution

It provides the first detailed three-term asymptotic expansion with optimal error bounds for the rightmost eigenvalue of non-Hermitian random matrices, establishing universality.

Findings

Three-term asymptotic expansion derived

Optimal error estimate established

Universality of all expansion terms proven

Abstract

We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an $n\times n$ random matrix with independent identically distributed complex entries as $n$ tends to infinity. All terms in the expansion are universal.