
Contents
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Abstract Abstract
-
23.1 Introduction 23.1 Introduction
-
23.2 Definition of random banded matrix (RBM) ensembles 23.2 Definition of random banded matrix (RBM) ensembles
-
23.3 Density of states 23.3 Density of states
-
23.3.1 Finite volume estimates on DOS 23.3.1 Finite volume estimates on DOS
-
23.3.2 Perturbation theory 23.3.2 Perturbation theory
-
-
23.4 Statistical mechanics and RBM 23.4 Statistical mechanics and RBM
-
23.5 Eigenvectors of RBM 23.5 Eigenvectors of RBM
-
23.5.1 Localization 23.5.1 Localization
-
23.5.2 Extended states 23.5.2 Extended states
-
23.5.3 More perturbation theory and scaling theory 23.5.3 More perturbation theory and scaling theory
-
23.5.4 Power law RBM 23.5.4 Power law RBM
-
-
23.6 Random sparse matrices 23.6 Random sparse matrices
-
23.7 Random Schrödinger on the Bethe lattice 23.7 Random Schrödinger on the Bethe lattice
-
Acknowledgments Acknowledgments
-
References References
-
-
-
-
-
-
23 Random banded and sparse matrices
Get accessThomas Spencer, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA, [email protected]
-
Published:08 August 2018
Cite
Abstract
This article discusses some mathematical results and conjectures about random band matrix ensembles (RBM) and sparse matrix ensembles. Spectral problems of RBM and sparse matrices can be expressed in terms of supersymmetric (SUSY) statistical mechanics that provides a dual representation for disordered quantum systems. This representation offers important insights into nonperturbative aspects of the spectrum and eigenfunctions of RBM. The article first presents the definition of RBM ensembles before considering the density of states, the behaviour of eigenvectors, and eigenvalue statistics for RBM and sparse random matrices. In particular, it highlights the relations with random Schrödinger (RS) and the role of the dimension of the lattice. It also describes the connection between RBM and statistical mechanics, the spectral theory of large random sparse matrices, conjectures and theorems about eigenvectors and local spacing statistics, and the RS operator on the Cayley tree or Bethe lattice.
Sign in
Personal account
- Sign in with email/username & password
- Get email alerts
- Save searches
- Purchase content
- Activate your purchase/trial code
- Add your ORCID iD
Purchase
Our books are available by subscription or purchase to libraries and institutions.
Purchasing informationMonth: | Total Views: |
---|---|
October 2022 | 9 |
December 2022 | 9 |
February 2023 | 2 |
April 2023 | 2 |
June 2023 | 1 |
July 2023 | 7 |
September 2023 | 8 |
October 2023 | 1 |
November 2023 | 7 |
January 2024 | 3 |
February 2024 | 4 |
March 2024 | 3 |
June 2024 | 10 |
July 2024 | 6 |
September 2024 | 3 |
November 2024 | 3 |
December 2024 | 2 |
February 2025 | 3 |
March 2025 | 5 |
April 2025 | 2 |
Get help with access
Institutional access
Access to content on Oxford Academic is often provided through institutional subscriptions and purchases. If you are a member of an institution with an active account, you may be able to access content in one of the following ways:
IP based access
Typically, access is provided across an institutional network to a range of IP addresses. This authentication occurs automatically, and it is not possible to sign out of an IP authenticated account.
Sign in through your institution
Choose this option to get remote access when outside your institution. Shibboleth/Open Athens technology is used to provide single sign-on between your institution’s website and Oxford Academic.
If your institution is not listed or you cannot sign in to your institution’s website, please contact your librarian or administrator.
Sign in with a library card
Enter your library card number to sign in. If you cannot sign in, please contact your librarian.
Society Members
Society member access to a journal is achieved in one of the following ways:
Sign in through society site
Many societies offer single sign-on between the society website and Oxford Academic. If you see ‘Sign in through society site’ in the sign in pane within a journal:
If you do not have a society account or have forgotten your username or password, please contact your society.
Sign in using a personal account
Some societies use Oxford Academic personal accounts to provide access to their members. See below.
Personal account
A personal account can be used to get email alerts, save searches, purchase content, and activate subscriptions.
Some societies use Oxford Academic personal accounts to provide access to their members.
Viewing your signed in accounts
Click the account icon in the top right to:
Signed in but can't access content
Oxford Academic is home to a wide variety of products. The institutional subscription may not cover the content that you are trying to access. If you believe you should have access to that content, please contact your librarian.
Institutional account management
For librarians and administrators, your personal account also provides access to institutional account management. Here you will find options to view and activate subscriptions, manage institutional settings and access options, access usage statistics, and more.