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Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices

Giorgio Cipolloni, László Endős, Dominik Schröder

Abstract

We prove that the mesoscopic linear statistics $\sum_i f(n^a(σ_i-z_0))$ of the eigenvalues $\{σ_i\}_i$ of large $n\times n$ non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any $H^{2}_0$-functions $f$ around any point $z_0$ in the bulk of the spectrum on any mesoscopic scale $0<a<1/2$. This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, $a=0$, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of $X$ at spectral parameters $z_1, z_2$ with an improved error term in the entire mesoscopic regime $|z_1-z_2|\gg n^{-1/2}$. The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator.

Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices

Abstract

We prove that the mesoscopic linear statistics of the eigenvalues of large non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any -functions around any point in the bulk of the spectrum on any mesoscopic scale . This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, , to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of at spectral parameters with an improved error term in the entire mesoscopic regime . The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator.
Paper Structure (14 sections, 20 theorems, 194 equations)

This paper contains 14 sections, 20 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof strategy
  4. Mesoscopic CLT for linear statistics: Proof of Theorem \ref{['theo:mesoclt']}
  5. Proof of Theorem \ref{['theo:impll']} for matrices with a Gaussian component
  6. GFT: Proof of Theorem \ref{['theo:impll']}
  7. Additional technical results
  8. Proof of Proposition \ref{['prop:CLTresm']}
  9. Asymptotic independence
  10. Proof of Lemma \ref{['pro:imppro']}
  11. Proof of Lemma \ref{['lem:mder']}
  12. Proof of Lemma \ref{['lem:newlem1']}
  13. Proof of Lemma \ref{['lem:m12bound']}
  14. Eigendecomposition of the stability operator

Key Result

Theorem 2.1

Let $X$ be an $n\times n$ matrix satisfying Assumption ass:1, fix a small $\tau>0$, and let $|z_0|\le 1-\tau$ and $a\in (0, \frac{1}{2})$. Let $f_{z_0,a}$ be defined as in eq:resctestf, with $f\in H_0^{2}(\Omega)$ for some a compact set $\Omega\subset \mathbf{C}$. Then $L_n(f_{z_0,a})$ converges (in with expectation $\mathop{\mathrm{\mathbf{E}}}\nolimits L(f)=0$, and second moments $\mathop{\mathr

Theorems & Definitions (41)

  • Remark 1.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3: Gaussian Free Field
  • Remark 2.4
  • Remark 2.5
  • Theorem 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Proposition 3.4: CLT for resolvents
  • ...and 31 more