The Perturbative Resolvent Method: spectral densities of random matrix ensembles via perturbation theory

Wenping Cui; Jason W. Rocks; Pankaj Mehta

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arXiv · 2012.00663

The Perturbative Resolvent Method: spectral densities of random matrix ensembles via perturbation theory

Wenping Cui·Jason W. Rocks·Pankaj Mehta

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Abstract

We present a simple, perturbative approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit we call the Perturbative Resolvent Method (PRM). The PRM is based on constructing a linear system of equations and calculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-circle Law for symmetric matrices, the Marchenko-Pastur Law for Wishart matrices, the spectral density for a product Wishart matrix composed of two square matrices, and the Circle and elliptic laws for real random matrices.

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Wenping Cui, Jason W. Rocks, Pankaj Mehta. 2020-12-01. The Perturbative Resolvent Method: spectral densities of random matrix ensembles via perturbation theory. https://arxiv.org/abs/2012.00663

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