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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 $b\in \mathbb{N}$ 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 $β$, 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 $β\equiv β_{n}$ 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 $β_{n}=\widehatβ (b/s)^{n/2}$ for $\widehatβ>0$, the normalized partition function of the system converges weakly as $n \to \infty$ to a distribution $\mathbf{L}(\widehatβ)$ depending continuously on the parameter $\widehatβ$. In the case $b=s$ we find a critical point in the behavior of the model when the inverse temperature is scaled as $β_{n}=\widehatβ/n$; for an explicitly computable critical value $κ_{b} > 0$ the variance of the normalized partition function converges to zero with large $n$ when $\widehatβ\leq κ_{b}$ and grows without bound when $\widehatβ>κ_{b}$. Finally, we prove a central limit theorem for the normalized partition function when $\widehatβ\leq κ_{b}$.