High-temperature scaling limit for directed polymers on a hierarchical lattice with bond disorder

TL;DR

This paper investigates the high-temperature limit of directed polymers on hierarchical diamond lattices with bond disorder, proving convergence of moments of the partition function in a specific scaling regime.

Contribution

It introduces a new scaling regime where the temperature increases with the graph layers and proves moment convergence of the partition function in this regime.

Findings

All positive integer moments of the partition function converge in the limit.

The proposed regime suggests a law of large numbers for the partition function.

The work extends previous results from the $b<s$ case to the $b=s$ case.

Abstract

Diamond "lattices" are sequences of recursively-defined graphs that provide a network of directed pathways between two fixed root nodes, $A$ and $B$. The construction recipe for diamond graphs depends on a branching number $b\in \mathbb{N}$ and a segmenting number $s\in \mathbb{N}$, for which a larger value of the ratio $s/b$ intuitively corresponds to more opportunities for intersections between two randomly chosen paths. By attaching i.i.d. random variables to the bonds of the graphs, I construct a random Gibbs measure on the set of directed paths by assigning each path an "energy" given by summing the random variables along the path. For the case $b=s$, I propose a scaling regime in which the temperature grows along with the number of hierarchical layers of the graphs, and the partition function (the normalization factor of the Gibbs measure) appears to converge in law. I prove that all of the positive integer moments of the partition function converge in this limiting regime. The motivation of this work is to prove a functional limit theorem that is analogous to a previous result obtained in the $b<s$ case.