Non trivial overlap distributions at zero temperature

We explore the consequences of Replica Symmetry Breaking at zero temperatur e. We introduce a repulsive coupling between a system and its unperturbed g round state. In the Replica Symmetry Breaking scenario a finite coupling in duces a non trivial overlap probability distribution among the unperturbed ground state and thr: one in presence of the coupling. We find a closed for mula for this probability for arbitrary ultrametric trees, in tt rms of the parameters defining the tree. The same probability is computed in numerica l simulations of a simple model with many ground states, but no ultrametric ity: polymers in random media in 1 + 1 dimension. This gives us an idea of what violation of our formula can be expected in cases when ultrametricity does not hold.