Continuum directed random polymers on disordered hierarchical diamond lattices

Jeremy Clark

Published 2018-02-11Version 1

I discuss a family of models for a continuum directed random polymer in a disordered environment in which the polymer lives on a fractal, $D^{b,s}$, called the \textit{diamond hierarchical lattice}. The diamond hierarchical lattice is a compact, self-similar metric space forming a network of interweaving pathways (continuum polymers) connecting a beginning node, $A$, to a termination node, $B$. This fractal depends on a branching parameter $b\in \{2,3,\cdots\}$ and a segmenting number $s\in \{2,3,\cdots\}$, and there is a canonical uniform probability measure $\mu$ on the collection of directed paths, $\Gamma^{b,s}$, for which the intersection set of two randomly chosen paths is almost surely either finite or of Hausdorff dimension $(\log s -\log b)/\log s$ when $s\geq b$. In the case $s>b$, my focus is on random measures on the set of directed paths that can be formulated as a subcritcal Gaussian multiplicative chaos measure with expectation $\mu$. When normalized, this random path measure is analogous to the continuum directed random polymer (CDRP) introduced by Alberts, Khanin, Quastel [Journal of Statistical Physics 154, 305-326 (2014)], which is formally related to the stochastic heat equation for a $(1+1)$-dimension polymer.