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, we define valleys by saying that two configurations submitted to the same stochastic forces belong to the same valley at time I iftheir trajectories have met before time t . We compute in the long-time limit the probability distribution of the number and the sizes of these valleys. We 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, we obtain non-selfaveraging effects for the number of macroscopic clusters and for their sizes. There are two kinds of dynamics one can consider to describe the time evolution of systems in statistical mechanics models: deterministic and stochastic dynamics. Deterministic dynamics are defined by a map F in phase space % , + I = F ( % , ) (1) which gives the configuration of a system at time t + 1 as a function of its configuration %, at time t. The map %, -+ V,,, does not depend on time. For such dynamics, phase space, even when it is finite, can be decomposed into several valleys, each valley being the basin of attraction of an attractor of the map F. Stochastic dynamics are defined by a map in phase space = G( %,, noise,) (2) which depends on some stochastic variables that we will call the noise, noise,. So the map V, + changes with time because it depends on noise, and it is only the statistical properties of this map (averages over noise,) which do not depend on time. In most cases, the random variable noise, represents the thermal noise and the map (2) allows paths from any configuration to any other configuration in phase space. The definition of valleys is much more difficult with stochastic dynamics. For finite systems, the valley structure depends on the timescale: in the limit t + 00, the system is able to explore the whole phase space and therefore one observes a single valley. On the contrary, for large systems and at finite t, one expects rather well defined valleys corresponding to the possible phases of the system. So one expects the number and the size of the valleys to depend on the timescale. In the present work, we will consider a very simplified model of stochastic dynamics: the annealed random map model which is defined by the following rules. (1) Phase space consists of M points. 0305-4470/88/090509 + 07$02.50 @ 1988 IOP Publishing Ltd L509 L510 Letter to the Editor (2) At each time step t, the map Vf + G( (ef, noise,) is a random map of this set of (3) The maps at time t and t' are uncorrelated. The valley structure of the quenched (deterministic) version of this model for which the map %, + (e,+, is random but remains fixed at all times has been studied recently (Derrida and Flyvbjerg 1987a). Since the notion of valleys depends on time for stochastic dynamics, we will use the following definition of valleys: we submit two different initial configurations (e, and (eh to the same noise, noise,, and we say that they belong to the same valley at time t if (e, = (e;. (Of course if two configurations meet at some time t, they remain identical at any later time.) This definition was already used numerically in several problems (Derrida and Weisbuch 1987) and gave well defined dynamical transitions. For the annealed random map model, to submit two configurations to the same noise, noise,, means simply that the same map (2) is used for the time evolution of and Vi. The goal of the present work is to calculate the statistical properties of the sizes of these valleys analytically. Let us start with the simple case of two randomly chosen configurations %, and (eh. If one defines A , the probability that (e, = (e; and B, the probability that (e, # %;, one has M points into itself (G( (e, noise,) and G( (e', noise,) are uncorelated if (e # (e'). 1 M A,,, = A , +B, B,+, = ( 1 -$) Bf with the initial condition A , = 0 and B-, = 1. This gives r + l 1 A , = B , = ( l d ) , (3)