Eigenstate thermalisation hypothesis and Gaussian fluctuations for Wigner matrices
Laszlo Erdos (Institute of Science and Technology Austria)
We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix W with an optimal error inversely proportional to the square root of the dimension. This verifies a strong form of Quantum Unique Ergodicity with an optimal convergence rate and we also prove Gaussian fluctuations around this convergence. The key technical tool is a new multi-resolvent local law for Wigner ensemble and the Dyson Brownian motion for eigenvector overlaps.