The intermediate disorder regime for a directed polymer model on a hierarchical lattice

Abstract

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number b ∈ N and a segment number s ∈ N. When b ≤ s previous work [27] has established that the model exhibits strong disorder for all positive values of the inverse temperature β, and thus weak disorder reigns only for β = 0 (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature β ≡ β n vanishes at an appropriate rate as the size n of the system grows. Our analysis requires separate treatment for the cases b < s and b = s. In the case b < s we prove that when the inverse temperature is taken to be of the form β n = β(b/s) n/2 for β > 0, the normalized partition function of the system converges weakly as n → ∞ to a distribution L( β) depending continuously on the parameter β. In the case b = s we find a critical point in the behavior of the model when the inverse temperature is scaled as β n = β/n; for an explicitly computable critical value κ b > 0 the variance of the normalized partition function converges to zero with large n when β ≤ κ b and grows without bound when β > κ b . Finally, we prove a central limit theorem for the normalized partition function when β ≤ κ b .

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

“…Let us mention some related work on the directed polymer model on the hierarchical lattice. 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.…”

Continuum directed random polymers on disordered hierarchical diamond lattices

“…This limit result is analogous to the intermediate disorder regime in [2] for directed polymers on the (1 + 1)-rectangular lattice. My focus here will be on developing the theory for a CDRP corresponding to the limiting partition function obtained in [1].…”

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

“…Although rectangular lattices are the most mathematically compelling graphical structures for studying directed polymers in disordered enviornments-due, in part, to their limiting connection with the stochastic heat equation-, it is also interesting to explore analogous models on graphical structures that have contrasting characteristics, such as exact hierarchical symmetry. The diamond hierarchical lattice is one such toy structure that researchers have chosen to grow their understanding of disordered polymers [10,19,2] and a variety of other statistical mechanical phenomena such as pinning models [14,20], resister networks [21,17,15], diffusion on fractals [18], and spin models [16]. Diamond hierarchical lattices are sequences D b,s n n∈N of recursively-defined finite graphs whose construction depends on a branching parameter b ∈ {2, 3, · · · } and a segmenting parameter s ∈ {2, 3, · · · } (see next section for details).…”

“…Thus, the respective cases s > b, s = b, and s < b are analogous to d < 2, d = 2, and d > 2 of the rectangular lattice, and this correspondence can be understood heuristically based on the expected number of sites shared by two randomly chosen directed paths; see the introduction of [19]. In [2] we studied an infinite-temperature distributional scaling limit analogous to [3] for the partition function of the diamond lattice polymer in the case when s > b and disorder variables are placed on either sites or bonds. The techniques of [2] do not extend to proving a limit theorem for the partition function in the s = b case, where it is relatively tricky to determine a plausible choice of infinite-temperature scaling β −1 ≡ β b n −1 ∞ as the generation, n, of the diamond graph D b,b n grows.…”

“…In [2] we studied an infinite-temperature distributional scaling limit analogous to [3] for the partition function of the diamond lattice polymer in the case when s > b and disorder variables are placed on either sites or bonds. The techniques of [2] do not extend to proving a limit theorem for the partition function in the s = b case, where it is relatively tricky to determine a plausible choice of infinite-temperature scaling β −1 ≡ β b n −1 ∞ as the generation, n, of the diamond graph D b,b n grows. In this article I propose an infinite-temperature scaling for the s = b diamond lattice polymer in the case of bond disorder.…”