The intermediate disorder regime for a directed polymer model on a hierarchical lattice

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number b∈N and a segment number s∈N. When b≤s it is known that the model exhibits strong disorder for all positive values of the inverse temperature β, and thus weak disorder reigns only for β=0 (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature β≡βn vanishes at an appropriate rate as the size n of the system grows. Our analysis requires separate treatment for the cases b0, the normalized partition function of the system converges weakly as n→∞ to a distribution L(β̂) and does so universally with respect to the initial weight distribution. We prove the convergence using renormalization group type ideas rather than the standard Wiener chaos analysis. In the case b=s we find a critical point in the behavior of the model when the inverse temperature is scaled as βn=β̂/n; for an explicitly computable critical value κb>0 the variance of the normalized partition function converges to zero with large n when β̂≤κb and grows without bound when β̂>κb. Finally, we prove a central limit theorem for the normalized partition function when β̂≤κb.