The conditional Gaussian multiplicative chaos structure underlying a critical continuum random polymer model on a diamond fractal

We discuss a Gaussian multiplicative chaos (GMC) structure underlying a family of random measures $\mathbf{M}_r$, indexed by $r\in\mathbb{R}$, on a space $\Gamma$ of directed pathways crossing a diamond fractal with Hausdorff dimension two. The laws of these random continuum path measures arise in a critical weak-disorder limiting regime for discrete directed polymers on disordered hierarchical graphs. For the analogous subcritical continuum polymer model in which the diamond fractal has Hausdorff dimension less than two, the random path measures can be constructed as subcritical GMCs through couplings to a spatial Gaussian white noise. This construction fails in the critical dimension two where, formally, an infinite coupling strength to the environmental noise would be required to generate the disorder. We prove, however, that there is a conditional GMC interrelationship between the random measures $(\mathbf{M}_r)_{r\in \mathbb{R}}$ such that the law of $\mathbf{M}_r$ can be constructed as a subcritical GMC with random reference measure $\mathbf{M}_R$ for any choice of $R\in (-\infty, r)$. A similar GMC structure plausibly would hold for a critical continuum (2+1)-dimensional directed polymer model.