Maximum of the characteristic polynomial of i.i.d. 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 Lambert et al. Electron. J. Probab. 29 (2024); the complex Ginibre case was covered in Lambert, Comm. Math Phys. 378 (2020). 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 (BRW) via Dyson Brownian motion. In particular, we find a new connection between real i.i.d. matrices and inhomogeneous BRW.