Eigenstate Thermalization Hypothesis for Generalized Wigner Matrices

Abstract: In this paper, we extend results of Eigenvector Thermalization to the case of generalized Wigner matrices. Analytically, the central quantity of interest here are multiresolvent traces, such as $\Lambda_A:= \frac{1}{N} \text{Tr }{ GAGA}$. In the case of Wigner matrices, as in \cite{cipolloni-erdos-schroder-2021}, one can form a self-consistent equation for a single $\Lambda_A$. There are multiple difficulties extending this logic to the case of general covariances. The correlation structure prevents us from deriving a self-consistent equation for a single matrix $A$; this is due to the introduction of new terms that are quite distinct from the form of $\Lambda_A$. We find a way around this by carefully splitting these new terms and writing them as sums of $\Lambda_B$, for matrices $B$ obtained by modifying $A$ using the covariance matrix. The result is a system of self-consistent equations relating families of deterministic matrices. Our main effort in this work is to derive and analyze this system of self-consistent equations.