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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 $bin mathbb{N}$ and a segmenting number $sin mathbb{N}$. 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)].
We construct and study a family of continuum random polymer measures $mathbf{M}_{r}$ corresponding to limiting partition function laws recently derived in a weak-coupling regime of polymer models on hierarchical graphs with marginally relevant disord
We study sums of directed paths on a hierarchical lattice where each bond has either a positive or negative sign with a probability $p$. Such path sums $J$ have been used to model interference effects by hopping electrons in the strongly localized re
We study the directed polymer model for general graphs (beyond $mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder, in terms of
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 $bin mathbb{N}$ and a segment
We consider two models for directed polymers in space-time independent random media (the OConnell-Yor semi-discrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic an