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

Abstract

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 $0<\kappa <\mathrm{\infty }$ such that as $n\to \mathrm{\infty }$ the variance of the normalized partition function tends to zero for $\hat{\beta }\le \kappa$ and grows without bound for $\hat{\beta }>\kappa$. 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 $0<{\alpha }_n\ll 1/\sqrt{n}$ and analyzing the asymptotic behavior of the variance. We show that when ${\alpha }_n=\alpha (\log n-\log \log n)/n^{3/2}$ (with a small modification to deal with non-zero third moment), there is a similar cutoff value $\eta$ for the parameter $\alpha$ such that the variance goes to zero when $\alpha <\eta$ and grows without bound when $\alpha >\eta$. Extending the analysis yet again by probing around the inverse temperature $$$(\kappa /\sqrt{n})+\eta (\log n-\log \log n)/n^{3/2}$, we find an infinite sequence of nested critical points for the variance behavior of the normalized partition function. In the subcritical cases $\hat{\beta }\le \kappa$ and $\alpha \le \eta$, this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.