Abstract: We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalization Hypothesis (ETH), which is a strong form of Quantum Unique Ergodicity (QUE) with optimal speed of convergence. Then, using this a priori bound as an input, we analyze the stochastic Eigenstate Equation (SEE) and prove Gaussian fluctuations in QUE.
The main methods behind the above results are: (i) multi-resolvents local laws established via a novel bootstrap scheme; (ii) energy estimates for SEE.