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 $0 < \kappa < \infty$ such that as $n \to \infty$ the variance of the normalized partition function tends to zero for $\hat{\beta} \leq \kappa$ 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 $\hat{\beta} \leq \kappa$ and $\alpha \leq \eta$, this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.

Publication
J. Theoret. Probab.
Tom Alberts
Tom Alberts
Associate Professor of Mathematics
University of Utah
Jeremy Clark
Jeremy Clark
Associate Professor of Mathematics
University of Mississippi

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