Continuum directed random polymers on disordered hierarchical diamond lattices

TL;DR

This paper studies a continuum directed random polymer model on a fractal lattice called the diamond hierarchical lattice, analyzing its properties using Gaussian multiplicative chaos in a subcritical regime.

Contribution

It introduces a new model of continuum directed polymers on a fractal lattice and explores its measure-theoretic properties using Gaussian multiplicative chaos techniques.

Findings

Establishes the existence of a subcritical Gaussian multiplicative chaos measure on the lattice.

Draws parallels between the fractal lattice model and the continuum directed polymers in Euclidean spaces.

Provides a framework for analyzing polymers on self-similar fractal structures.

Abstract

I discuss models for a continuum directed random polymer in a disordered environment in which the polymer lives on a fractal called the \textit{diamond hierarchical lattice}, a self-similar metric space forming a network of interweaving pathways. This fractal depends on a branching parameter $b\in \mathbb{N}$ and a segmenting number $s\in \mathbb{N}$. For $s>b$ my focus is on random measures on the set of directed paths that can be formulated as a subcritical Gaussian multiplicative chaos. This path measure is analogous to the continuum directed random polymer introduced by Alberts, Khanin, Quastel [Journal of Statistical Physics \textbf{154}, 305-326 (2014)].