Directed Polymers on Hierarchical Lattices with site disorder

TL;DR

This paper investigates a hierarchical lattice polymer model with site disorder, establishing free energy existence, conditions for strong disorder, and analyzing high-temperature behavior and fluctuations.

Contribution

It introduces new conditions for very strong disorder in hierarchical lattice polymers and provides detailed analysis of free energy and fluctuation bounds.

Findings

Existence of free energy for the model.

Necessary and sufficient conditions for very strong disorder.

Bounds on the fluctuation exponent of log Z_n.

Abstract

We study a polymer model on hierarchical lattices very close to the one introduced and studied in \cite{DGr, CD}. For this model, we prove the existence of free energy and derive the necessary and sufficient condition for which very strong disorder holds for all $\gb$, and give some accurate results on the behavior of the free energy at high-temperature. We obtain these results by using a combination of fractional moment method and change of measure over the environment to obtain an upper bound, and second moment method to get a lower bound. We also get lower bounds on the fluctuation exponent of $\log Z_n$, and study the infinite polymer measure in the weak disorder phase.