Statistical properties of valleys in the annealed random map model

The annealed random map model is one of the simplest models of statistical mechanics with stochastic dynamics. For this model, the authors define valleys by saying that two configurations submitted to the same stochastic forces belong to the same valley at time t if their trajectories have met before time t. They compute in the long-time limit the probability distribution of the number and the sizes of these valleys. They find a structure very reminiscent of the valley structure of the mean-field spin glasses with sample-to-sample fluctuations. Interpreting the annealed random map model as an aggregation model, they obtain non-self-averaging effects for the number of macroscopic clusters and for their sizes.