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 ${\widehat{\beta }}/\sqrt{n}$ for a parameter ${\widehat{\beta }}>0$, we show that there is a cutoff value $0< \kappa < \infty$ such that as $n \rightarrow \infty$ the variance of the normalized partition function tends to zero for ${\widehat{\beta }}\le \kappa$ and grows without bound for ${\widehat{\beta }}> \kappa$. 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 ${\widehat{\beta }}\le \kappa$ and $\alpha \le \eta$, this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.