A random energy model for size dependence: recurrence vs. transience

We investigate the size dependence of disordered spin models having an infinite number of Gibbs measures in the framework of a simplified 'random energy model for size dependence'. We introduce two versions (involving either independent random walks or branching processes), that can be seen as generalizations of Derrida's random energy model. Our model is shown to exhibit a recurrence/transience transition for Gibbs measures (meaning: a.s. existence/nonexistence of subsequences of volumes converging to a given random infinite volume state), depending on the growthrate of a function MN describing the 'number of extremal Gibbs states that can be observed in a finite volume N'. We investigate the model in detail in the 'critical regime' M-N similar to (logN)(p) (With critical point p = 1). We obtain the a.s. large volume asymptotics of the relative weights for finding a particular state and we compute the set of a.s. cluster points of the corresponding occupation times (corresponding to the 'empirical metastate'). In the course of the proof we obtain Laws of the Iterated Logarithm for the pair X-N(v), Y-N(v) : sup(mu = 1,2,...,MN;mu not equal v) X-N(mu) where (X-N(mu)) (mu = 1,...,MN;N = 1,2,....) is either a branching random walk or a collection of random walks, independent over mu.