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

Abstract

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 $b\in \mathbb{N}$ and a segment number $s\in \mathbb{N}$. When $b\leq s$ previous work [27] has established that the model exhibits strong disorder for all positive values of the inverse temperature $\beta$, and thus weak disorder reigns only for $\beta=0$ (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature $\beta\equiv \beta_{n}$ vanishes at an appropriate rate as the size $n$ of the system grows. Our analysis requires separate treatment for the cases $b<s$ and $b=s$. In the case $b<s$ we prove that when the inverse temperature is taken to be of the form $\beta_{n}=\widehat{\beta} (b/s)^{n/2}$ for $\widehat{\beta}>0$, the normalized partition function of the system converges weakly as $n \to \infty$ to a distribution $\mathbf{L}(\widehat{\beta})$ depending continuously on the parameter $\widehat{\beta}$. In the case $b=s$ we find a critical point in the behavior of the model when the inverse temperature is scaled as $\beta_{n}=\widehat{\beta}/n$; for an explicitly computable critical value $\kappa_{b} > 0$ the variance of the normalized partition function converges to zero with large $n$ when $\widehat{\beta}\leq \kappa_{b}$ and grows without bound when $\widehat{\beta}>\kappa_{b}$. Finally, we prove a central limit theorem for the normalized partition function when $\widehat{\beta}\leq \kappa_{b}$.