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Random matrices: Universality of local spectral statistics of non-Hermitian matrices

Terence Tao, Van Vu

Abstract

It is a result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on $\mathbb{C}$ with kernel $K_{\infty}(z,w):=\frac{1}πe^{-|z|^2/2-|w|^2/2+z\bar{w}}$ in the limit $n\to\infty$. We show that this asymptotic law is universal among all random $n\times n$ matrices $M_n$ whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles. These are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this, we work with the log-determinants $\log|\det(M_n-z_0)|$ rather than with the Stieltjes transform $\frac{1}{n}\operatorname {tr}(M_n-z_0)^{-1}$. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real $n\times n$ matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has $\sqrt{\frac{2n}π}+o(\sqrt{n})$ real eigenvalues asymptotically almost surely.

Random matrices: Universality of local spectral statistics of non-Hermitian matrices

Abstract

It is a result of Ginibre that the normalized bulk -point correlation functions of a complex Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on with kernel in the limit . We show that this asymptotic law is universal among all random matrices whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles. These are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this, we work with the log-determinants rather than with the Stieltjes transform . Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has real eigenvalues asymptotically almost surely.

Paper Structure

This paper contains 24 sections, 41 theorems, 469 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The real case and applications
  3. Key ideas and a sketch of the proof
  4. Key propositions
  5. Notation
  6. A concentration inequality
  7. From log-determinant concentration to the local circular law
  8. Reduction to the Four Moment Theorem and log-determinant concentration
  9. The complex case
  10. The real case
  11. Quick applications
  12. Resolvent swapping
  13. Proof of the Four Moment Theorem
  14. Concentration of log-determinant for gaussian matrices
  15. The upper bound
  16. ...and 9 more sections

Key Result

Theorem \oldthetheorem

Let $M_n, \tilde{M}_n$ be independent-entry matrix ensembles with independent real and imaginary parts, obeying Condition C1, such that $M_n$ and $\tilde{M}_n$ both match moments with the complex gaussian matrix ensemble to third order, and match moments with each other to fourth order. Let $k \geq for some fixed $m$ and some smooth functions $F_{i,j}: {\mathbb{C}} \to {\mathbb{C}}$ for $i=1,\ldo

Figures (4)

  • Figure 1: The cumulative distribution function for the number of eigenvalues in the disk $B(0,\sqrt{n}/3)$ of real gaussian and real Bernoulli matrices of size $10,000 \times 10,000$, after normalizing the mean by $n/9$ and variance by $\sqrt{n}$. Thanks to Ke Wang for the data and figure.
  • Figure 2: The spectrum of a random real gaussian $10,000 \times 10,000$ matrix, with additional detail near the origin to show the concentration on the real axis. Thanks to Ke Wang for the data and figure.
  • Figure 3: The spectrum of a random real Bernoulli $10,000 \times 10,000$ matrix, with additional detail near the origin. Thanks to Ke Wang for the data and figure.
  • Figure 4: The empirical average number of real eigenvalues of $200$ samples of real gaussian and real Bernoulli matrices of various sizes, plotted against $\sqrt{\frac{2n}{\pi}}$. Thanks to Ke Wang for the data and figure.

Theorems & Definitions (94)

  • Definition \oldthetheorem: Independent-entry matrices
  • Theorem \oldthetheorem: Four Moment Theorem for complex matrices
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem: Asymptotic kernel
  • Lemma \oldthetheorem: Kernel asymptotics
  • proof
  • Corollary \oldthetheorem: Universality for complex matrices
  • proof
  • Remark \oldthetheorem
  • ...and 84 more