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An elementary mean-field approach to the spectral densities of random matrix ensembles

Author

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  • Cui, Wenping
  • Rocks, Jason W.
  • Mehta, Pankaj

Abstract

We present a simple mean-field approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit. Our approach 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.

Suggested Citation

  • Cui, Wenping & Rocks, Jason W. & Mehta, Pankaj, 2024. "An elementary mean-field approach to the spectral densities of random matrix ensembles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 637(C).
    • Handle: RePEc:eee:phsmap:v:637:y:2024:i:c:s037843712400116x
    DOI: 10.1016/j.physa.2024.129608
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    References listed on IDEAS

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    1. Joseph W. Baron & Tobias Galla, 2020. "Dispersal-induced instability in complex ecosystems," Nature Communications, Nature, vol. 11(1), pages 1-9, December.
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