Anomalous rate of eigenstate thermalisation at singularities of the density of states

We prove the Eigenstate Thermalisation Hypothesis (ETH), also known as Quantum Unique Ergodicity (QUE), for large $N\times N$ mean-field random matrices with general correlation structure. We identify the microcanonical ensemble and establish the optimal fluctuation scale of eigenvector overlaps around it. Our results invalidate the prediction by Feingold and Peres [Phys. Rev. A 34, 591 (1986)] in the physics literature of quantum chaos, based upon popular semiclassical theory, and uncover the genuine mechanism which relies on multi-resolvent local laws. Although fluctuations are expected to increase as the density of states vanishes, and indeed scale as $N^{-1/2}$ in the special cusp regime, rather than $N^{-1}$ in the bulk, we find, unexpectedly, that the same $N^{-1}$ rate persists at regular spectral edges. Hence, generically, in the absence of cusps, the entire eigenbasis fluctuates on the same scale as a Haar unitary. This anomaly stems from delicate cancellations in the solution of the underlying matrix Dyson equation, which form the core of our analysis.