Strong quantum unique ergodicity and its Gaussian fluctuations for Wigner matrices

Abstract:   We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation 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 analyse the Stochastic Eigenstate Equation (SEE) and prove the Gaussian fluctuations in the QUE. The main methods behind these results are: (i) multi-resolvents local laws established via a novel bootstrap scheme; (ii) energy estimates for SEE.