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Fluctuation Around the Circular Law for Random Matrices with Real Entries

Giorgio Cipolloni, László Erdős, Dominik Schröder

Abstract

We extend our recent result [Cipolloni, Erdős, Schröder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices $X$ with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [O'Rourke, Renfrew 2016] or the first four moments of the matrix elements match the real Gaussian [Tao, Vu 2015; Kopel 2015]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [Cipolloni, Erdős, Schröder 2019] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.

Fluctuation Around the Circular Law for Random Matrices with Real Entries

Abstract

We extend our recent result [Cipolloni, Erdős, Schröder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [O'Rourke, Renfrew 2016] or the first four moments of the matrix elements match the real Gaussian [Tao, Vu 2015; Kopel 2015]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [Cipolloni, Erdős, Schröder 2019] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.

Paper Structure

This paper contains 27 sections, 24 theorems, 327 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Connection to the Gaussian free field
  4. Universality of the local singular value statistics of X-z close to zero
  5. Proof strategy
  6. Central limit theorem for linear statistics: Proof of Theorem \ref{['theo:CLT']}
  7. Local law away from the imaginary axis: Proof of Theorem \ref{['theo:Gll']}
  8. CLT for resolvents: Proof of Proposition \ref{['prop clt resolvent']}
  9. Asymptotic independence of resolvents: Proof of Proposition \ref{['prop:indmr']}
  10. Overview of the proof of Proposition \ref{['prop:indeig']}
  11. Comparison of l with the regularised process l
  12. Definition of the partially correlated processes lu
  13. Proof of Proposition \ref{['prop:indeig']}
  14. Relations among the scales in the proof of Proposition \ref{['prop:indeig']}
  15. Universality and independence of the singular values of X-z1,X-z2 close to zero: Proof of Theorems \ref{['theo:un']} and \ref{['theo:indun']}
  16. ...and 12 more sections

Key Result

Theorem 2.2

Let $X$ be a real $n\times n$ i.i.d. matrix satisfying Assumption ass:1 with eigenvalues $\{\sigma_i\}_{i=1}^n$, and denote the fourth cumulantNote that in the real case the fourth cumulant is given by $\kappa_4=\kappa(\chi,\chi,\chi,\chi)=\mathop{\mathrm{\mathbf{E}}}\nolimits\chi^4-3$, while in the to complex Gaussian random variables $L(f)$ with expectation $\mathop{\mathrm{\mathbf{E}}}\nolimits

Figures (1)

  • Figure 1: Proof overview for Proposition \ref{['prop:indeig']}: The collections of eigenvalues $\bm\lambda^{z_l}$ of $H^{z_l}$ for different $l$'s are approximated by several stochastic processes. The processes ${\bm \mu}=\bm\mu^{(l)}$ are independent for different $l$'s by definition.

Theorems & Definitions (51)

  • Theorem 2.2: Central Limit Theorem for linear statistics
  • Remark 2.3
  • Remark 2.4: Comparison with 1510.02987 and MR3540493
  • Remark 2.5: Comparison with the complex case
  • Remark 2.6: Real correction to the expected circular law
  • Remark 2.7: Special case: Polynomial test functions
  • Theorem 2.8: Universality of small singular values of $X-z$
  • Remark 2.9
  • Theorem 2.10: Asymptotic independence of small singular values of $X-z_1, X-z_2$
  • Remark 2.11
  • ...and 41 more