Weak-Disorder Limit at Criticality for Directed Polymers on Hierarchical Graphs

Abstract

We prove a distributional limit theorem conjectured in Clark (J Stat Phys 174(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.