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 $\mu$.