Weak-disorder limit for directed polymers on critical hierarchical graphs with vertex disorder

TL;DR

This paper investigates the behavior of directed polymers on hierarchical graphs with vertex-based disorder, establishing a limit theorem for the partition function in the critical case where the graph's branching and segmentation numbers are equal.

Contribution

It extends previous models by analyzing vertex disorder in the critical case, providing a distributional limit theorem for the partition function as the system grows.

Findings

Proves a distributional limit theorem for the partition function.

Focuses on the critical case where branching number equals segmentation number.

Extends prior work from edge-based to vertex-based disorder models.

Abstract

We study models for a directed polymer in a random environment (DPRE) in which the polymer traverses a hierarchical diamond graph and the random environment is defined through random variables attached to the vertices. For these models, we prove a distributional limit theorem for the partition function in a limiting regime wherein the system grows as the coupling of the polymer to the random environment is appropriately attenuated. The sequence of diamond graphs is determined by a choice of a branching number $b\in \{2,3,\ldots\}$ and segmenting number $s\in \{2,3,\ldots\}$, and our focus is on the critical case of the model where $b=s$. This extends recent work in the critical case of analogous models with disorder variables placed at the edges of the graphs rather than the vertices.