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

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 joint scaling which number layers, \(n\in {\mathbb {N}}\), grows as inverse temperature, \(\beta \equiv \beta (n)\), vanishes fine-tuned dependence n. conjecture pertains to marginally relevant case model wherein branching parameter \(b \in \{2,3,\ldots \}\) and segmenting \(s determining are equal, coincides fractal embedding having Hausdorff dimension two. Unlike analogous weak-disorder random polymer \(b<s\) (or (1+1)-dimensional rectangular lattice), convergence function when \(b=s\) cannot be approached through term-by-term Wiener chaos expansion, does not exist continuum emerging limit. analysis proceeds by controlling terms Wasserstein distance perturbative generalization Stein’s method critical step.