  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.
