A framework maps Boltzmann-weighted lattice configurations to correlated random matrix ensembles via real-space to momentum-space variance profiles, deriving spectral moments and resolvent densities benchmarked on Ising and Edwards-Anderson models.
The matrix Dyson equation and its applications for random matrices
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abstract
These lecture notes are a concise introduction of recent techniques to prove local spectral universality for a large class of random matrices. The general strategy is presented following the recent book with H.T. Yau. We extend the scope of this book by focusing on new techniques developed to deal with generalizations of Wigner matrices that allow for non-identically distributed entries and even for correlated entries. This requires to analyze a system of nonlinear equations, or more generally a nonlinear matrix equation called the Matrix Dyson Equation (MDE). We demonstrate that stability properties of the MDE play a central role in random matrix theory. The analysis of MDE is based upon joint works with J. Alt, O. Ajanki, D. Schr\"oder and T. Kr\"uger that are supported by the ERC Advanced Grant, RANMAT 338804 of the European Research Council. The lecture notes were written for the 27th Annual PCMI Summer Session on Random Matrices held in 2017. The current edited version will appear in the IAS/Park City Mathematics Series, Vol. 26.
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2026 3verdicts
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The spectral weak-recovery threshold for linearized AMP in the multi-view spiked Wigner model is SNR(λ,B)=1, where SNR is the largest eigenvalue of Diag(√λ)(B⊙B)Diag(√λ), and this coincides with the information-theoretic threshold for a broad class of spike priors.
In the linear-width regime, the second GD step yields a spiked random matrix whose number of outliers is floor(alpha2 / (1/2 - alpha1)), and batch reuse enables learning directions with information exponent greater than one under suitable alpha scalings.
citing papers explorer
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Random Matrix Spectra from Boltzmann-Weighted Lattice Ensembles
A framework maps Boltzmann-weighted lattice configurations to correlated random matrix ensembles via real-space to momentum-space variance profiles, deriving spectral moments and resolvent densities benchmarked on Ising and Edwards-Anderson models.
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Sharp Spectral Thresholds for Multi-View Spiked Wigner Models
The spectral weak-recovery threshold for linearized AMP in the multi-view spiked Wigner model is SNR(λ,B)=1, where SNR is the largest eigenvalue of Diag(√λ)(B⊙B)Diag(√λ), and this coincides with the information-theoretic threshold for a broad class of spike priors.
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Feature Learning in Linear-Width Two-Layer Networks: Two vs. One Step of Gradient Descent
In the linear-width regime, the second GD step yields a spiked random matrix whose number of outliers is floor(alpha2 / (1/2 - alpha1)), and batch reuse enables learning directions with information exponent greater than one under suitable alpha scalings.