Maximum of the Characteristic Polynomial of I.I.D. Matrices

TL;DR

This paper determines the asymptotic behavior of the maximum of the characteristic polynomial for i.i.d. matrices, including real and complex cases, using coupling with branching random walks and Dyson Brownian motion.

Contribution

It provides the first universality results for the non-Hermitian case of the Fyodorov–Hiary–Keating conjecture, especially for real Ginibre matrices.

Findings

Asymptotic maximum for characteristic polynomial computed

New connection established between i.i.d. matrices and branching random walk

Results extend to real and complex Ginibre matrices

Abstract

We compute the leading order asymptotic of the maximum of the characteristic polynomial for i.i.d. matrices with real or complex entries. In particular, this result is new even for real Ginibre matrices, which was left as an open problem in [arXiv:2303.09912]; the complex Ginibre case was covered in [arXiv:1902.01983]. These are the first universality results for the non--Hermitian analog of the first order term of the Fyodorov--Hiary--Keating conjecture. Our methods are based on constructing a coupling to the branching random walk via Dyson Brownian motion. In particular, we find a new connection between real i.i.d. matrices and inhomogeneous branching random walk.