Electronic Journal of Probability

The phase transition in the ultrametric ensemble and local stability of Dyson Brownian motion

Per von Soosten and Simone Warzel

Full-text: Open access

Abstract

We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N} $, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are based on the flow of the resolvents of matrices with a random diagonal component under Dyson Brownian motion, for which we establish submicroscopic stability results for short times. These results go beyond norm-based continuity arguments for Dyson Brownian motion and complement the existing analysis after the local equilibration time.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 70, 24 pp.

Dates
Received: 18 January 2018
Accepted: 10 July 2018
First available in Project Euclid: 26 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1532570598

Digital Object Identifier
doi:10.1214/18-EJP197

Mathematical Reviews number (MathSciNet)
MR3835476

Zentralblatt MATH identifier
06924682

Subjects
Primary: 15A52 47B80: Random operators [See also 47H40, 60H25]

Keywords
Dyson Brownian motion localization transition local statistics ultrametric ensemble

Rights
Creative Commons Attribution 4.0 International License.

Citation

von Soosten, Per; Warzel, Simone. The phase transition in the ultrametric ensemble and local stability of Dyson Brownian motion. Electron. J. Probab. 23 (2018), paper no. 70, 24 pp. doi:10.1214/18-EJP197. https://projecteuclid.org/euclid.ejp/1532570598


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