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Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

László Erdős, Torben Krüger, Dominik Schröder

Abstract

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also used in the companion paper [arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven.

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

Abstract

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also used in the companion paper [arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven.

Paper Structure

This paper contains 25 sections, 30 theorems, 261 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The Dyson equation
  4. Cusp universality
  5. Local law
  6. Local Law
  7. Stability and shape analysis
  8. Probabilistic bound
  9. Bootstrapping
  10. Rigidity and absence of eigenvalues
  11. Cusp fluctuation averaging and proof of Theorem \ref{['thm pfD bound moments']}
  12. Graphical representation via double index graphs
  13. Single index graphs
  14. Single index resolution
  15. Examples of graphs
  16. ...and 10 more sections

Key Result

Theorem 2.3

Let $H$ be a complex Hermitian Wigner matrix satisfying Assumptions bdd moments--bdd m. Assume that the self-consistent density $\rho$ within $[\tau-\kappa,\tau+\kappa]$ from Assumption bdd m has a physical cusp, i.e. that $\rho$ is locally given by gamma def eqs for some $\gamma>0$ and $\rho$ eithe where $\bm x=(x_1,\dots,x_k)$, $\mathop{\mathrm{\mathstrut d}} \nolimits^{}

Figures (2)

  • Figure 1: Figure \ref{['rhos deltas plot']} illustrates the evolution of $\rho^\text{fc}_s$ along the semicircular flow at two times $0<s<t_\ast<s'$ before and after the cusp. We recall that $\rho^\ast=\rho^\text{fc}_0$ and $\rho=\rho^\text{fc}_{t_\ast}$. Figure \ref{['rhos xi plot']} shows the points $\xi_s(\mathfrak{e}_s^\pm)$ as well as their distances to the edges $\mathfrak{e}_0^\pm$.
  • Figure 2: Representative cusp analysis. Figures \ref{['small_scale_phase']} and \ref{['large_scale_phase']} show the level set $\Re g(z)=0$. On a small scale $g(z)\sim z^4$, while on a large scale $g(z)\sim z^2$. Figure \ref{['gammas_plot']} shows the final deformed and rescaled contours $\widehat{\Upsilon}'$ and $\widehat{\Gamma}'$. Figure \ref{['small_scale_phase']} furthermore shows the cone sections $\Omega_k^>$ and $\Omega_k^{<}$, where we for clarity do not indicate the precise area thresholds given by $\delta$ and $R$. We also do not specifically indicate $\Omega_k^<$ for $k = \pm 1,\pm 2,\pm 3$ as then $\mathop{\mathrm{cc}}\nolimits(\Omega_k^<)=\mathop{\mathrm{cc}}\nolimits(\Omega_k^>)$, cf. Claims 4--5 in the proof of Lemma \ref{['contour deform lemma']}.

Theorems & Definitions (67)

  • Definition 2.1: Self-consistent density of states
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4: Fluctuation scale
  • Theorem 2.5: Local law
  • Corollary 2.6: Uniform rigidity
  • Corollary 2.7: Cusp rigidity
  • Corollary 2.8: No eigenvalues outside the support of the self-consistent density
  • Remark 2.9
  • Remark 3.1
  • ...and 57 more