  This paper is the last of the series investigating renormalization group
aspects of stochastic random matrices, including a Wigner-like disorder. We
consider the equilibrium dynamics formalism that can be merged with the Ward
identities arising from the large N effective kinetics. We construct a
regulator that does not break time-reversal symmetry and show that the
resulting flow equations reduce to the equilibrium flow built in our previous
works. Finally, we investigate the flow equations beyond the equilibrium
dynamics assumption and study the stability of the perturbation around the
fluctuation-dissipation theorem.
