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Law of fractional logarithm for random matrices

Zhigang Bao, Giorgio Cipolloni, László Erdős, Joscha Henheik, Oleksii Kolupaiev

Abstract

We prove the Paquette-Zeitouni law of fractional logarithm (LFL) for the extreme eigenvalues [arXiv:1505.05627] in full generality, and thereby verify a conjecture from [arXiv:1505.05627]. Our result holds for any Wigner minor process and both symmetry classes, in particular for the GOE minor process, while [arXiv:1505.05627] and the recent full resolution of LFL by Baslingker et.~al.~[arXiv:2410.11836] cover only the GUE case which is determinantal. Lacking the possibility for a direct comparison with the Gaussian case, we develop a robust and natural method for both key parts of the proof. On one hand, we rely on a powerful martingale technique to describe precisely the strong correlation between the largest eigenvalue of an $N\times N$ Wigner matrix and its $(N-k)\times (N-k)$ minor if $k\ll N^{2/3}$. On the other hand, we use dynamical methods to show that this correlation is weak if $k\gg N^{2/3}$.

Law of fractional logarithm for random matrices

Abstract

We prove the Paquette-Zeitouni law of fractional logarithm (LFL) for the extreme eigenvalues [arXiv:1505.05627] in full generality, and thereby verify a conjecture from [arXiv:1505.05627]. Our result holds for any Wigner minor process and both symmetry classes, in particular for the GOE minor process, while [arXiv:1505.05627] and the recent full resolution of LFL by Baslingker et.~al.~[arXiv:2410.11836] cover only the GUE case which is determinantal. Lacking the possibility for a direct comparison with the Gaussian case, we develop a robust and natural method for both key parts of the proof. On one hand, we rely on a powerful martingale technique to describe precisely the strong correlation between the largest eigenvalue of an Wigner matrix and its minor if . On the other hand, we use dynamical methods to show that this correlation is weak if .

Paper Structure

This paper contains 16 sections, 10 theorems, 126 equations.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. Notations
  4. Proof of Theorem \ref{['thm:main']}
  5. Decorrelation estimate: Lower bound for $\limsup$ and upper bound for $\liminf$
  6. Lower bound for $\limsup$
  7. Upper bound for $\liminf$
  8. Martingale inequalities: Upper bound for $\limsup$ and lower bound for $\liminf$
  9. Upper bound for $\limsup$
  10. Lower bound for $\liminf$
  11. Concluding the proof
  12. Decorrelation estimate: Proof of Proposition \ref{['prop:decor']}
  13. Dyson Brownian motion: Proof of Proposition \ref{['prop:DBM']}
  14. Green function comparison: Proof of Proposition \ref{['prop:GFT']}
  15. Proof of Corollary \ref{['cor:limits']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $(\lambda_1^{(N)})_{N \in \mathbf{N}}$ be the (shifted and scaled according to eq:TWconv) largest eigenvalues of a real symmetric ($\beta = 1$) or complex Hermitian ($\beta = 2$) Wigner minor process $(H^{(N)})_{N \in \mathbf{N}}$ as in Definition def:minor. Then, almost surely, we have

Theorems & Definitions (18)

  • Definition 1.1: Wigner minor process
  • Theorem 1.2: Law of fractional logarithm
  • Corollary 1.3: All possible limit points
  • Remark 1.4: Extensions beyond Wigner matrices
  • Proposition 2.1: Small deviation tail estimates
  • proof : Proof of Proposition \ref{['prop:tails']}
  • Proposition 2.2: Decorrelation estimate on tail events
  • Lemma 2.3: Extension lemma
  • proof : Proof of Lemma \ref{['lem:ext']}
  • Proposition 3.1: Step 1: Decorrelation for Gaussian divisible ensemble
  • ...and 8 more