Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

Abstract: We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and $β$-ensembles. In the special case of Gaussian-divisible Wigner matrices, our method provides universality of the maximum up to tightness. These are the first universal results on the Fyodorov--Hiary--Keating conjectures for these models, and in particular answer the question of optimal rigidity for the spectrum of Wigner matrices. Our proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlated fields and Gaussian multiplicative chaos.

• The paper demonstrates that the maximum of the log‐characteristic polynomial scales as fβ log N with explicit universal constants (f1 = √2, f2 = 1).
• It employs innovative stochastic dynamics and moment matching methods to compare general Wigner matrices with Gaussian-divisible ensembles.
• The study establishes new optimal eigenvalue rigidity estimates with sub-Gaussian decay, confirming sharp constants on the (log N)/N scale.

Optimal Rigidity and Maxima of the Characteristic Polynomial in Wigner Matrices

Introduction and Context

The paper "Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices" (2312.13335) provides a comprehensive analysis of two central questions in random matrix theory: the extremal behavior of the characteristic polynomial for a broad class of Wigner matrices and general $\beta$-ensembles, and the asymptotic behavior of the maximum deviation of eigenvalues from their deterministic classical location—often termed eigenvalue rigidity. The work is significant, especially for extending universality results around the Fyodorov-Hiary-Keating (FHK) program beyond integrable systems to encompass real symmetric and complex Hermitian Wigner matrices with general entry distributions.

Main Results

Leading Order for Maximum of the Log-Characteristic Polynomial

The authors establish that, for a Wigner matrix $H$ of size $N \times N$ (either real symmetric or complex Hermitian), the centered maximum of the real part of the logarithm of the characteristic polynomial on the bulk interval $I = [A+\epsilon, B-\epsilon]$ admits the leading order asymptotics: $$\sup_{E \in I} \operatorname{Re} \left[ \log|\det(E - H)| - \mathbb{E}\log|\det(E - H)| \right] = f_{\beta} \log N + o(\log N) \quad \text{w.h.p.}$$ where $f_\beta$ is a universal constant depending on the symmetry class: $f_1 = \sqrt{2}$ for real symmetric (GOE), $f_2 = 1$ for complex Hermitian (GUE).

For Gaussian-divisible ensembles (Wigner matrices plus a small GOE component), the first order is precise, and the extremal statistics can be coupled tightly with the corresponding Gaussian ensemble.

The proof methodology is entirely distinct from determinantal or Riemann–Hilbert approaches previously dominant in the GUE/CUE literature; it combines stochastic dynamics (Dyson Brownian motion), local laws for eigenvalue linear statistics, and techniques from the theory of log-correlated Gaussian fields and Gaussian multiplicative chaos (GMC).

Universality and the FHK Conjecture

The results provide the first universal (non-integrable) confirmation of the leading-order form of the FHK conjecture for the maximum of the log-characteristic polynomial in the bulk regime for Wigner and $\beta$-ensembles, previously established only for a handful of integrable models [(2312.13335), Sec 1.1]. Specifically, they prove that both the precise leading order and the correct scale of fluctuations are universal.

Optimal Eigenvalue Rigidity

A second major contribution is a new set of optimal rigidity estimates for Wigner and $\beta$-ensembles, both in the typical ($1 - o(1)$) and overwhelming probability ($1 - N^{-D}$ for all $D$) regimes. For eigenvalues $\lambda_k$ and classical locations $\gamma_k$ set by the limiting measure, the authors prove

$$\max_{k \in [\alpha N, (1-\alpha)N]} |\lambda_k - \gamma_k| = \sqrt{2} \frac{\sqrt{\log N}}{N \rho(\gamma_k)}(1 + o(1))$$

with overwhelming probability, for a precise density $\rho$ at location $\gamma_k$. Additionally, they establish sub-Gaussian decay in the large deviation regime, extending the known scales of rigidity for both Wigner matrices and general log-gas ($\beta$-ensemble) particle systems.

This is the first time such precision for the maximal deviation (with optimal multiplicative constants and on the $(\log N)/N$ scale) has been established for non-unitary invariant models. Previously, only suboptimal scales—such as $(\log N)^{C}/N$—were known.

Technical and Methodological Innovations

1. Stochastic Dynamics Approach: The paper uses dynamics-based comparison (Dyson Brownian motion) between complex ensembles and their Gaussian analogues, tracking the evolution of spectra and characteristic polynomials through matrix-valued SDEs.
2. Comparison and Moment Matching: By leveraging four-moment matching and quantitative comparison theorems (extending techniques initiated by Tao and Vu, Landon et al.), the authors transfer universality results at the level of Gaussian-divisible matrices to arbitrary Wigner matrices. The analysis includes careful controls of large moments and extremely exceptional events, which are crucial for overwhelming probability estimates.
3. GMC and Log-Correlated Fields: The approach to lower bounds (i.e., proving the correct order of the supremum) is via connecting the log-characteristic polynomial to GMC, using Laplace transforms of linear statistics and intricate stochastic analysis akin to the treatment of extremal processes in the two-dimensional Gaussian free field.
4. Extension to $\beta$-ensembles: Through adaptations of Johansson's loop equations and strong rigidity results under non-quadratic potentials, the paper generalizes all main results to one-cut, analytic-potential, repulsive log-gas ensembles.

Numerical and Theoretical Claims

• The order of the maximum is explicitly computed, with a sharp constant.
• For Gaussian-divisible Wigner matrices, the maximum of the log-characteristic polynomial is shown to be universal up to tightness; that is, all fluctuations on scales smaller than $O(1)$ are common between the general class and the GOE/GUE.
• Rigidity: The fluctuation of the maximal deviation is sub-Gaussian, i.e., probabilities decay as $\exp(-c u^2)$ for a deviation $u\gg \sqrt{\log N}/N$, and precise constants in these exponents are given.
• These are the first results to extend FHK universality for maxima and optimal rigidity with constants to all Wigner and log-gas ensembles.

Implications and Future Directions

The formalism developed in this work provides a robust template for universality proofs for spectral extremes, potentially applicable to several outstanding questions. Notably:

• Non-Hermitian Matrices: The extension of these results to non-Hermitian random matrices with independent entries is within reach, with work appearing based on Fourier-analytic methods [cf. (Cipolloni et al., 2024)].
• Graph Ensembles & Free Probability: The coupling and comparison techniques could be adapted to random graphs, adjacency matrices, and broader classes in free probability.
• Higher Order/Edge Asymptotics: The paper does not address higher-order corrections or the spectral edge, pointing to future work where the interplay of dynamics and rigidity will be further explored.
• Connections to Extreme Value Theory and Quantum Chaos: The methods highlight the deep connections between the extremal properties of random spectral measures and those of log-correlated Gaussian fields, with direct relevance for understanding "freezing transitions" and universality in quantum chaotic systems and number theory.

Conclusion

This work provides a rigorous probabilistic framework for analyzing extremal observables of Wigner and $\beta$-ensembles, establishing universal leading-order results for maxima of the log-characteristic polynomial and optimal eigenvalue rigidity. The dynamical coupling and comparison method developed here mark a substantial advancement in treating non-integrable random matrix models, with broad implications for extreme value statistics, the universality paradigm, and applications reaching from mathematical physics to number theory.