Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The eigenvalues of i.i.d. matrices are hyperuniform

Giorgio Cipolloni, László Erdős, Oleksii Kolupaiev

Abstract

We prove that the point process of the eigenvalues of real or complex non-Hermitian matrices $X$ with independent, identically distributed entries is hyperuniform: the variance of the number of eigenvalues in a subdomain $Ω$ of the spectrum is much smaller than the volume of $Ω$. Our main technical novelty is a very precise computation of the covariance between the resolvents of the Hermitization of $X-z_1, X-z_2$, for two distinct complex parameters $z_1,z_2$.

The eigenvalues of i.i.d. matrices are hyperuniform

Abstract

We prove that the point process of the eigenvalues of real or complex non-Hermitian matrices with independent, identically distributed entries is hyperuniform: the variance of the number of eigenvalues in a subdomain of the spectrum is much smaller than the volume of . Our main technical novelty is a very precise computation of the covariance between the resolvents of the Hermitization of , for two distinct complex parameters .
Paper Structure (56 sections, 45 theorems, 604 equations, 1 figure)

This paper contains 56 sections, 45 theorems, 604 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof strategy
  4. Proof of Theorem \ref{['theo:main1']}
  5. Ideas behind the proofs of Propositions \ref{['prop:E_omega']} and \ref{['prop:Var_main']}
  6. Mesoscopic regime: $\eta\in [\eta_c,T]$
  7. Sub-microscopic regime: $\eta\in [\eta_L,\eta_0]$
  8. Critical regime: $\eta\in [\eta_0,\eta_c]$
  9. The real case
  10. Local laws: set-up and results
  11. Single-resolvent local law
  12. Novel averaged two-resolvent local law
  13. Multi-resolvent local laws for longer chains
  14. Calculations with the Girko's formula
  15. Preliminary reductions in the Girko's formula
  16. ...and 41 more sections

Key Result

Theorem 1.1

Let $X$ be an i.i.d. matrix and consider any nice domain $\Omega\subset\mathbf{D}$. Then, there exists a $q>0$ such that

Figures (1)

  • Figure 1: In gray, we illustrated a macroscopic ($\alpha=0$) domain $\Omega_N$ in the case when it intersects the real line. The dashed lines correspond to $\Im z=\pm d_0$, where $d_0$ is given by Assumption \ref{['ass:Omega_real']}. In this example, the intersection of $\partial\Omega_N$ and the set $|\Im z|\le d_0$ consists of two curves transversely intersecting the real axis. Finally, the scattered dots depict the eigenvalues of a large real i.i.d. matrix $X$.

Theorems & Definitions (66)

  • Theorem 1.1: Informal statement
  • Remark 2.2
  • Theorem 2.4: Complex case
  • Theorem 2.6: Real case
  • Remark 2.7
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3: Portmanteau principle
  • Proposition 3.4
  • Proposition 3.5: Left tail of the smallest singular value distribution
  • ...and 56 more