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On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime

Published in Springer

2022

Volume: 187

Issue: 3

We study the local eigenvalue statistics ξNω,E${\xi}_{\omega ,E}^{N}$ associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)$(2N+1)\times (2N+1)$ random band matrices with independent, identically distributed, real random variables and band width growing as Nα${N}^{\alpha}$, for 0<α<12$0<\alpha <\frac{1}{2}$. We consider the limit points associated with the random variables ξNω,E[I]${\xi}_{\omega ,E}^{N}[I]$, for I⊂R$I\subset \mathbb{R}$, and E∈(−2,2)$E\in (-2,2)$. For random band matrices with Gaussian distributed random variables and for 0≤α<17$0\le \alpha <\frac{1}{7}$, we prove that this family of random variables has nontrivial limit points for almost every E∈(−2,2)$E\in (-2,2)$, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables ξNω,E[I]${\xi}_{\omega ,E}^{N}[I]$ and associated quantities related to the intensities, as *N* tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for 0<α<12$0<\alpha <\frac{1}{2}$, we prove that any nontrivial limit points of the random variables ξNω,E[I]${\xi}_{\omega ,E}^{N}[I]$ are distributed according to Poisson distributions.We study the local eigenvalue statistics ξNω,E${\xi}_{\omega ,E}^{N}$ associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)$(2N+1)\times (2N+1)$ random band matrices with independent, identically distributed, real random variables and band width growing as Nα${N}^{\alpha}$, for 0<α<12$0<\alpha <\frac{1}{2}$. We consider the limit points associated with the random variables ξNω,E[I]${\xi}_{\omega ,E}^{N}[I]$, for I⊂R$I\subset \mathbb{R}$, and E∈(−2,2)$E\in (-2,2)$. For random band matrices with Gaussian distributed random variables and for 0≤α<17$0\le \alpha <\frac{1}{7}$, we prove that this family of random variables has nontrivial limit points for almost every E∈(−2,2)$E\in (-2,2)$, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables ξNω,E[I]${\xi}_{\omega ,E}^{N}[I]$ and associated quantities related to the intensities, as *N* tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for 0<α<12$0<\alpha <\frac{1}{2}$, we prove that any nontrivial limit points of the random variables ξNω,E[I]${\xi}_{\omega ,E}^{N}[I]$ are distributed according to Poisson distributions.

Postprint Version

Content may be subject to copyright.This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Spr... ...This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s10955-022-02923-5

About the journal

Journal | Data powered by TypesetJournal of Statistical Physics |
---|---|

Publisher | Data powered by TypesetSpringer |

Open Access | Yes |