High-Temperature Scaling Limit for Directed Polymers on a Hierarchical Lattice with Bond Disorder

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 ∈ N and a segmenting number s ∈ 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.

On the Moments of the $$(2+1)$$-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window

“…In particular, for the marginally relevant case, Alberts, Clark and Kocić in [ACK17] established the existence of a phase transition, similar to [CSZ17a]. And more recently, Clark [Cla17] computed the moments of the partition function around a critical window for the case of bond disorder. The computations in the hierarchical lattice case employ the independence structure inherent in hierarchical models, which is not available on Z d .…”

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

“…This section begins by defining a family of random measures on directed paths crossing diamond hierarchical graphs and concludes with the statement of Theorem 2.7, which was conjectured in [10] and is the main result of this article. A directed path on D b,s n is a function p : {1, .…”

“…b (r) grows super-factorially with m ∈ N. Thus the following theorem was left as a conjecture in [10].…”

“…The main result of the current article is a proof of an analogous limit theorem for W ω n (β) in the b = s case. The correct choice of inverse temperature scaling-see below in (2.3)-was introduced in [10] although the results therein were confined to proving convergence of the positive integer moments. Convergence of the integer moments does not imply convergence in law because the higher limiting moments increase super-factorially; see (III) of Theorem 2.4 below.…”

Continuum directed random polymers on disordered hierarchical diamond lattices

“…A similar CDRP model on the b = s diamond lattice, if it exists, will require a different approach for its construction; see [6] for computations relevant to the continuum limit of discrete polymers. It is interesting to compare this question with results [7,8] by Caravenna, Sun, and Zygouras on scaling limits of the partition function for (1 + 2)-rectangular lattice polymers.…”