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, rigorously verifying the Eigenstate Thermalization Hypothesis.
We prove a CLT for quadratic forms of eigenvectors of Wigner matrices with arbitrary deterministic matrices, considerably strengthening previous results on quantum unique ergodicity.
We show that already a small noise completely thermalizes the bulk singular vectors of arbitrary deterministic matrices. In particular we prove a strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence for such matrices.
