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

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∈{2,3,…} and segmenting number s∈{2,3,…}, 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.