Nested critical points for a directed polymer on a disordered diamond lattice

We consider a model for a directed polymer in a random environment defined on a hierarchical diamond lattice in which i.i.d. random variables are attached to the lattice bonds. Our focus is on scaling schemes in which a size parameter $n$, counting the number of hierarchical layers of the system, becomes large as the inverse temperature $\beta$ vanishes. When $\beta$ has the form $\hat{\beta}/\sqrt{n}$ for a parameter $\hat{\beta}>0$, we show that there is a cutoff value $0 \kappa $. We obtain a more refined description of the border between these two regimes by setting the inverse temperature to $\kappa/\sqrt{n} + \alpha_n$ where $0 \eta$. Extending the analysis yet again by probing around the inverse temperature $\kappa/\sqrt{n} + \eta \log n/n^2$ we find an infinite sequence of nested critical points for the variance behavior of the normalized partition function. In the subcritical cases $\hat{\beta} \leq \kappa$ and $\alpha \leq \eta$ this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.