On the statistical mechanics approach in the random matrix theory: Integrated density of states
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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.
Key WordsRandom matrix integrating density of states statistical mechanics mean field-theory
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