[Paper Review] Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

László Erdős, Torben Krüger+1 more|arXiv (Cornell University)|Sep 11, 2018

Random Matrices and Applications88 references40 citations

TL;DR

This paper establishes cusp universality for complex Hermitian Wigner-type random matrices by proving an optimal local law at the cusp, demonstrating that local eigenvalue statistics follow a Pearcey process. The result completes the Wigner-Dyson-Mehta universality conjecture for the complex Hermitian class, showing that cusp singularities in the density of states lead to universal statistics governed by the Pearcey kernel, even for approximate cusps with extended Pearcey processes.

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.

Motivation & Objective

• To resolve the final remaining case of the Wigner-Dyson-Mehta universality conjecture for complex Hermitian matrices by proving cusp universality.
• To establish that local eigenvalue statistics at cusp singularities of the limiting density of states are universal and described by the Pearcey kernel.
• To extend universality to approximate cusps, where an extended Pearcey process emerges.
• To provide the optimal local law at the cusp as a foundational technical result for the broader universality program.
• To complete the classification of universal local eigenvalue statistics in random matrix theory for the complex Hermitian symmetry class.

Proposed method

• Prove an optimal local law for the Green's function near cusp singularities of the density of states in complex Wigner-type matrices.
• Use a three-step strategy: (1) establish the local law at optimal scale, (2) prove universality for ensembles with a small Gaussian component, and (3) remove the Gaussian component via perturbation theory.
• Analyze the Dyson equation and its solution to classify possible singularities in the density of states, confirming that only square root, cubic root (cusp), and edge singularities occur.
• Employ spectral parameter-dependent estimates and high-probability bounds on the Green's function to control fluctuations near the cusp.
• Apply the local law to derive precise apriori bounds essential for the universality proof in steps two and three.
• Leverage the classification of singularities to show that cusp formation is the only mechanism for band merging in Wigner-type ensembles, justifying the universality of the Pearcey process.

Experimental results

Research questions

• RQ1 Do local eigenvalue statistics at cusp singularities of the density of states in complex Hermitian Wigner-type matrices follow a universal distribution?
• RQ2 Can the universality of the Pearcey process be established for general Wigner-type matrices without i.i.d. entries or matrix integral representations?
• RQ3 What happens to the local eigenvalue statistics when the cusp is not exact but approximate, i.e., when the density exhibits a near-cusp shape?
• RQ4 Is the optimal local law at the cusp sufficient to establish universality in the three-step strategy for WDM universality?
• RQ5 How does the emergence of the Pearcey kernel relate to the merging of spectral bands in one-parameter families of Wigner-type ensembles?

Key findings

• The local eigenvalue statistics at cusp singularities of the limiting density of states are universal and described by the Pearcey kernel, completing the Wigner-Dyson-Mehta universality conjecture for the complex Hermitian class.
• The optimal local law for the Green's function is proven at the cusp, with high-probability convergence down to the optimal scale near the spectral edge.
• For approximate cusps, the local eigenvalue statistics are governed by an extended Pearcey process, with the parameter determined by the ratio of the local lengthscale to the eigenvalue spacing.
• The cusp universality result is robust under general conditions on the matrix entries, including non-i.i.d. and non-Gaussian distributions.
• The local law at the cusp is the key technical input for the companion paper on real symmetric Wigner-type matrices and underpins recent results on non-Hermitian random matrices.
• The classification of singularities in the density of states confirms that only square root, cubic root (cusp), and edge singularities occur, with cusp singularities arising naturally from band merging in parameterized ensembles.