# Weak-disorder limit at criticality for directed polymers on hierarchical graphs

## TL;DR

This paper proves a distributional limit theorem for directed polymer partition functions on hierarchical graphs at criticality, revealing new behavior in the marginally relevant disorder case with joint scaling of layers and temperature.

## Contribution

It establishes the first distributional convergence result for the critical marginally relevant case of directed polymers on hierarchical graphs, using a novel Stein's method approach.

## Key findings

- Distributional convergence of partition functions at criticality.
- Limit theorem applies to models with edge and vertex disorder.
- Analysis introduces a perturbative Stein's method at a critical scale.

## Abstract

We prove a distributional limit theorem conjectured in [Journal of Statistical Physics 174, No. 6, 1372-1403 (2019)] for partition functions defining models of directed polymers on diamond hierarchical graphs with disorder variables placed at the graphical edges. The limiting regime involves a joint scaling in which the number of hierarchical layers, $n\in \mathbb{N}$, of the graphs grows as the inverse temperature, $\beta\equiv \beta(n)$, vanishes with a fine-tuned dependence on $n$. The conjecture pertains to the marginally relevant disorder case of the model wherein the branching parameter $b \in \{2,3,\ldots\}$ and the segmenting parameter $s \in \{2,3,\ldots\}$ determining the hierarchical graphs are equal, which coincides with the diamond fractal embedding the graphs having Hausdorff dimension two. Unlike the analogous weak-disorder scaling limit for random polymer models on hierarchical graphs in the disorder relevant $b<s$ case (or for the (1+1)-dimensional polymer on the rectangular lattice), the distributional convergence of the partition function when $b=s$ cannot be approached through a term-by-term convergence to a Wiener chaos expansion, which does not exist for the continuum model emerging in the limit. The analysis proceeds by controlling the distributional convergence of the partition functions in terms of the Wasserstein distance through a perturbative generalization of Stein's method at a critical step. In addition, we prove that a similar limit theorem holds for the analogous model with disorder variables placed at the vertices of the graphs.