Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality
Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality
复制标题
DOI：
10.1002/cpa.21895
复制
发表时间：
2020-04-19
影响因子：
3
通讯作者：
Yin, Jun
中科院分区：
文献类型：
--
作者：
Bourgade, Paul;Yau, Horng-Tzer;Yin, Jun
考虑对称的一维随机带阵，它具有任意元素的一般分布和带宽。在光谱的大部分和最大范围内，我们得到了以下结果。(I)半圆形定律对任何人都是适用的。(Ii)在任意固定能量下，本征值局部收敛于高斯正交系综所给出的点过程。(Iii)所有特征向量都是离域的，这意味着它们的范数都同时以压倒性的概率有界(归一化后)。(Iv)量子唯一遍历性成立，即本征向量的局域质量以压倒性概率变得均匀分布。我们将所需的平均场折减法\cite{BouErdYauYin2017}扩展到当前设置。本文的两个新观点是：(1)对带状矩阵的广义预解给出了一个新的估计。它的证明以及改进的波动平均估计将在本系列的第二部分和第三部分中介绍。(2)随机矩阵的量子唯一遍历性的强(高概率)形式。作为证明，我们构造了特征向量重叠的完美匹配观测，并证明了它们在矩阵布朗运动下满足特征向量矩流方程{BouYau2017}。
Considersymmetric one-dimensional random band matrices with general distribution of the entries and band widthfor any. In the bulk of the spectrum and in the largelimit, we obtain the following results. (i) The semicircle law holds up to the scalefor any. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning theirnorms are all simultaneously bounded by(after normalization in) with overwhelming probability, for any. (iv )Quantum unique ergodicity holds, in the sense that the localmass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method \cite{BouErdYauYin2017}, which required, to the current setting. Two new ideas are: (1) A new estimate on the "generalized resolvent" of band matrices when. Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series \cite {BouYanYauYin2018,YanYin2018}. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation \cite{BouYau2017} under the matrix Brownian motions.