This paper presents a novel approach to characterize the dynamics of the limit spectrum of large random matrices. This approach is based upon the notion we call spectral dominance. In particular, we show that the limit spectral measure can be determined as the derivative of the unique viscosity solution of a partial integro-differential equation. This also allows to make general and short proofs for the convergence problem. We treat the cases of Dyson Brownian motions, Wishart processes and present a general class of models for which this characterization holds.
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