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Spectral Statistics of Structured Random Matrices

Total Cost €


EC-Contrib. €






Project "RandMat" data sheet

The following table provides information about the project.


Organization address
city: GENEVE
postcode: 1211

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Switzerland [CH]
 Project website
 Total cost 1˙257˙442 €
 EC max contribution 1˙257˙442 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2016-STG
 Funding Scheme ERC-STG
 Starting year 2017
 Duration (year-month-day) from 2017-01-01   to  2021-12-31


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    UNIVERSITE DE GENEVE CH (GENEVE) coordinator 1˙257˙442.00


 Project objective

The purpose of this proposal is a better mathematical understanding of certain classes of large random matrices. Up to very recently, random matrix theory has been mainly focused on mean-field models with independent entries. In this proposal I instead consider random matrices that incorporate some nontrivial structure. I focus on two types of structured random matrices that arise naturally in important applications and lead to a rich mathematical behaviour: (1) random graphs with fixed degrees, such as random regular graphs, and (2) random band matrices, which constitute a good model of disordered quantum Hamiltonians.

The goals are strongly motivated by the applications to spectral graph theory and quantum chaos for (1) and to the physics of conductance in disordered media for (2). Specifically, I will work in the following directions. First, derive precise bounds on the locations of the extremal eigenvalues and the spectral gap, ultimately obtaining their limiting distributions. Second, characterize the spectral statistics in the bulk of the spectrum, using both eigenvalue correlation functions on small scales and linear eigenvalue statistics on intermediate mesoscopic scales. Third, prove the delocalization of eigenvectors and derive the distribution of their components. These results will address several of the most important questions about the structured random matrices (1) and (2), such as expansion properties of random graphs, hallmarks of quantum chaos in random regular graphs, crossovers in the eigenvalue statistics of disordered conductors, and quantum diffusion.

To achieve these goals I will combine tools introduced in my previous work, such as local resampling of graphs and subdiagram resummation techniques, and in addition develop novel, robust techniques to address the more challenging goals. I expect the output of this proposal to contribute significantly to the understanding of structured random matrices.


year authors and title journal last update
List of publications.
2019 Roland Bauerschmidt, Jiaoyang Huang, Antti Knowles, Horng-Tzer Yau
Edge rigidity and universality of random regular graphs of intermediate degree
published pages: , ISSN: , DOI:
2018 Yukun He, Matteo Marcozzi
Diffusion Profile for Random Band Matrices: a Short Proof
published pages: , ISSN: , DOI:
2017 Florent Benaych-Georges, Charles Bordenave, Antti Knowles
Largest eigenvalues of sparse inhomogeneous Erdős-Rényi graphs
published pages: , ISSN: , DOI:
2018 Yukun He
Mesoscopic linear statistics of Wigner matrices of mixed symmetry class
published pages: , ISSN: , DOI:
2017 Florent Benaych-Georges, Charles Bordenave, Antti Knowles
Spectral radii of sparse random matrices
published pages: , ISSN: , DOI:

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