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Universality of extremal eigenvalues of large random matrices

Giorgio Cipolloni, László Erdős, Yuanyuan Xu

Abstract

We prove that the spectral radius of a large random matrix $X$ with independent, identically distributed complex entries follows the Gumbel law irrespective of the distribution of the matrix elements. This solves a long-standing conjecture of Bordenave and Chafa{\"ı} and it establishes the first universality result for one of the most prominent extremal spectral statistics in random matrix theory. Furthermore, we also prove that the argument of the largest eigenvalue is uniform on the unit circle and that the extremal eigenvalues of $X$ form a Poisson point process.

Universality of extremal eigenvalues of large random matrices

Abstract

We prove that the spectral radius of a large random matrix with independent, identically distributed complex entries follows the Gumbel law irrespective of the distribution of the matrix elements. This solves a long-standing conjecture of Bordenave and Chafa{\"ı} and it establishes the first universality result for one of the most prominent extremal spectral statistics in random matrix theory. Furthermore, we also prove that the argument of the largest eigenvalue is uniform on the unit circle and that the extremal eigenvalues of form a Poisson point process.
Paper Structure (40 sections, 39 theorems, 486 equations, 1 figure)

This paper contains 40 sections, 39 theorems, 486 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main result
  3. Related universality results
  4. New ingredients
  5. Dynamical approach to multi-resolvent local laws
  6. Main results
  7. Spectral radius
  8. Rightmost eigenvalue
  9. Main technical results: Multi--resolvent local laws at the spectral edge of $X$
  10. Local law for $G^{z}$
  11. Local law for $G^{z_1}AG^{z_2}$ near the cusp
  12. Optimality of the $\langle G^{z_1}A_1G^{z_2}A_2\rangle$ local law
  13. Local law for $G^zF$ and $G^zFG^zF^*$ near the cusp
  14. Smallest singular value
  15. Proofs of Theorems \ref{['thm:gumbel']}, \ref{['thm:poisson']} and \ref{['thm:maxRe']}
  16. ...and 25 more sections

Key Result

Theorem 2.2

Let $X$ be a complex i.i.d. matrix satisfying Assumption ass:mainass. Set Then for any fixed $r\in {\mathbb R }$ and $\theta \in [0,2\pi)$, we haveWe formulated the convergence as a limiting statement, but inspecting our proof we can also obtain an effective speed of convergence $O((\log\log n)^2/(\log n))$ in theta_x_joint uniformly in $(r,\theta)$ similarly to maxRe_Gin

Figures (1)

  • Figure 1: For $z$ in the regime $|z|>1$ slightly beyond the unit disk, the picture on the right shows the corresponding density $\rho^z$ of $H^z$, which exhibits a small gap around zero of size $\Delta\sim (|z|^2-1)^{3/2}$; see the paragraph below AEK19.

Theorems & Definitions (81)

  • Theorem 2.2: Spectral radius
  • Remark 2.3
  • Remark 2.4: Convergence of moments
  • Theorem 2.5: Poisson Point Process
  • Theorem 2.6: Rightmost eigenvalue
  • Theorem 3.1: Theorem 3.1 in SpecRadius
  • Corollary 3.2: Corollary 3.2 in SpecRadius
  • Theorem 3.3: Local law with $|z_1-z_2|$-decorrelation
  • Corollary 3.4
  • proof
  • ...and 71 more