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

Giorgio Cipolloni, Benjamin Landon

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.

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 [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.
Paper Structure (46 sections, 66 theorems, 488 equations, 1 table)

This paper contains 46 sections, 66 theorems, 488 equations, 1 table.

Table of Contents

  1. Introduction
  2. Logarithmic correlated fields in random matrices
  3. Emergence of log--correlated fields in other models
  4. Methods
  5. Branching random walk structure and lower bound
  6. Upper bound
  7. Main result
  8. Preliminaries
  9. Fine rigidity estimates for the Hermitization of $X-z$
  10. Proof of Proposition \ref{['lem:precllaw']}
  11. Removal of Gaussian divisible component
  12. Local laws for matrices of mixed symmetry
  13. Maximum on almost-global scales
  14. Upper bound of $\Psi_n (z)$ for complex i.i.d. matrices with Gaussian component
  15. Proof of Lemma \ref{['lem:charup']}
  16. ...and 31 more sections

Key Result

Theorem 2.2

Let $X$ be a complex i.i.d. matrix as in Definition def:model. Then for any $\varepsilon >0$ and $0 < r < 1$ we have that

Theorems & Definitions (88)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4: Comparison with the complex case
  • Remark 2.5: Maximum over the real axis
  • Remark 2.6
  • Lemma 2.7
  • Theorem 2.8
  • Proposition 2.9
  • Definition 3.1
  • ...and 78 more