  We show that the trap model at its critical temperature presents dynamical
ultrametricity in the sense of Cugliandolo and Kurchan [CuKu94]. We use the
explicit analytic solution of this model to discuss several issues that arise
in the context of mean-field glassy dynamics, such as the scaling form of the
correlation function, and the finite time (or finite forcing) corrections to
ultrametricity, that are found to decay only logarithmically with the
associated time scale, as well as the fluctuation dissipation ratio. We also
argue that in the multilevel trap model, the short time dynamics is dominated
by the level which is at its critical temperature, so that dynamical
ultrametricity should hold in the whole glassy temperature range. We revisit
some experimental data on spin-glasses in light of these results.
