From Dyson to Pearcey: Universal statistics in random matrix theory

D.J. Schröder, From Dyson to Pearcey: Universal Statistics in Random Matrix Theory, IST Austria, 2019.

Download
OA 4.23 MB

Thesis | Published | English
Department
Series Title
IST Austria Thesis
Abstract
In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion. In the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal 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. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime. In the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure.
Publishing Year
Date Published
2019-03-18
Page
375
IST-REx-ID

Cite this

Schröder DJ. From Dyson to Pearcey: Universal Statistics in Random Matrix Theory. IST Austria; 2019. doi:10.15479/AT:ISTA:th6179
Schröder, D. J. (2019). From Dyson to Pearcey: Universal statistics in random matrix theory. IST Austria. https://doi.org/10.15479/AT:ISTA:th6179
Schröder, Dominik J. From Dyson to Pearcey: Universal Statistics in Random Matrix Theory. IST Austria, 2019. https://doi.org/10.15479/AT:ISTA:th6179.
D. J. Schröder, From Dyson to Pearcey: Universal statistics in random matrix theory. IST Austria, 2019.
Schröder DJ. 2019. From Dyson to Pearcey: Universal statistics in random matrix theory, IST Austria, 375p.
Schröder, Dominik J. From Dyson to Pearcey: Universal Statistics in Random Matrix Theory. IST Austria, 2019, doi:10.15479/AT:ISTA:th6179.
Main File(s)
File Name
Access Level
OA Open Access
Last Uploaded
2019-03-28T08:53:52Z

Source File
Access Level
Restricted Closed Access
Last Uploaded
2019-08-13T13:46:44Z

Export

Marked Publications

Open Data IST Research Explorer

Search this title in

Google Scholar