Wigner矩阵的本征态热化假设
Eigenstate Thermalization Hypothesis for Wigner Matrices
摘要：
我们证明了任何确定性矩阵近似是一个大的随机Wigner矩阵的特征基单位，具有很高的概率和最佳误差与维数的平方根成反比。因此，我们的定理严格验证了Deutsch [Deutsch 1991]对最简单的混沌量子系统维格纳系综的本征态热化假设。在数学方面，我们证明了量子唯一遍历性（QUE）的强形式，同时具有所有特征向量的最佳收敛速度，推广了[Bournut，Yau 2017]和[Bournut，Yau，Yin 2020]中以前的概率QUE结果。
英文摘要：
We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalization Hypothesis by Deutsch [Deutsch 1991] for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in [Bourgade, Yau 2017] and [Bourgade, Yau, Yin 2020].
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