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arXiv · 2302.00157

Eigenstate Thermalization Hypothesis for Generalized Wigner Matrices

Arka Adhikari·Sofiia Dubova·Changji Xu·Jun Yin
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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 $Λ_A:= \frac{1}{N} \text{Tr }{ GAGA}$. In the case of Wigner matrices, as in \cite{cipolloni-erdos-schroder-2021}, one can form a self-consistent equation for a single $Λ_A$. 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 $Λ_A$. We find a way around this by carefully splitting these new terms and writing them as sums of $Λ_B$, 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.

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Arka Adhikari, Sofiia Dubova, Changji Xu, Jun Yin. 2023-02-16. Eigenstate Thermalization Hypothesis for Generalized Wigner Matrices. https://arxiv.org/abs/2302.00157

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