Kuelske, C.
A Random Energy Model for Size Dependence:
Recurrence vs. Transience
(364K, PS)

ABSTRACT.  We investigate the size dependence of disordered mean field 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 $M_N$ 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' 
$M_N\sim \left(\log N\right)^p$ (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 $X_N^{\mu}, Y_N^{\nu} :=\sup_{{\mu=1,2,\dots,M_N} 
{\mu\neq \nu}}X_N^{\mu}$  where $\left(X_N^{\mu}\right)_{\mu=1,\dots,M_N; 
N=1,2,\dots}$ is either a branching random walk or a collection 
of random walks, independent over $\mu$.