Universality of local spectral statistics of random matrices
Universality of local spectral statistics of random matrices
复制标题
随机矩阵局部谱统计的普适性
DOI：
10.1090/s0273-0979-2012-01372-1
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
H. Yau
中科院分区：
文献类型：
--
作者：
L. Erdős;H. Yau
Wigner-Gaudin-Mehta-Dyson猜想认为大型随机矩阵的局部特征值统计量表现出仅依赖于矩阵系综的对称类的普适行为。对于不变矩阵模型，特征值分布由对数气体给出，逆温度$\beta = 1，2，4$，对应于正交，酉和辛系综。对于$\beta \not \in \{1，2，4\}$，在这个模型后面没有矩阵模型，但是对数气体的统计物理解释对于所有$\beta > 0$仍然有效。不变系综的普适性猜想断言局部特征值统计量与V无关。在这篇文章中，我们回顾了我们最近的解决方案的普适性猜想不变和非不变的合奏。我们还将证明戴森布朗运动的局部遍历性是普适性背后的内在机制。在此基础上，我们讨论了Dyson布朗运动的局部弛豫时间的Dyson猜想的解。相关的问题，如本征向量的离域化和局部版本的维格纳循环定律也将讨论。
The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given by a log-gas with inverse temperature $\beta = 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles. For $\beta \not \in \{1, 2, 4\}$, there is no matrix model behind this model, but the statistical physics interpretation of the log-gas is still valid for all $\beta > 0$. The universality conjecture for invariant ensembles asserts that the local eigenvalue statistics are independent of $V$. In this article, we review our recent solution to the universality conjecture for both invariant and non-invariant ensembles. We will also demonstrate that the local ergodicity of the Dyson Brownian motion is the intrinsic mechanism behind the universality. Furthermore, we review the solution of Dyson's conjecture on the local relaxation time of the Dyson Brownian motion. Related questions such as delocalization of eigenvectors and local version of Wigner's semicircle law will also be discussed.
影响因子：
3.1
作者：
Erdos, Laszlo;Schlein, Benjamin;Yau, Horng-Tzer
通讯作者：
Yau, Horng-Tzer