Abstract
We consider the ensemble of random symmetricn×n matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn→∞ and that this limiting distribution is the solution of a certain self-consistent equation.
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de Monvel, A.B., Pastur, L. & Shcherbina, M. On the statistical mechanics approach in the random matrix theory: Integrated density of states. J Stat Phys 79, 585–611 (1995). https://doi.org/10.1007/BF02184872
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DOI: https://doi.org/10.1007/BF02184872