The random map model: a disordered model with deterministic dynamics

The random map model is a simple disordered system with deterministic dynamics. For each point in phase space, one chooses at random another point in phase space as being its successor in time. Phase space is broken into basins of several attractors. We obtain the analytic expression for the probability distribution f(Ws) of the weights Ws, where Ws denotes the normalized size of the basin of the s-th attractor. We also compute the probability distribution π (Y) of Y where Y is defined by Y = Σ W2s . When we compare s f(W) and π (Y) in the random map model and in the mean field theory of spin glasses, we find that the shapes are very similar in both models but the analytic expressions are different.