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We construct and study a family random continuum polymer measures$\mathbf{M}_{r}$ corresponding to limiting partition function laws recentlyderived in a weak-coupling regime of polymer models on hierarchical graphs withmarginally relevant disorder. The continuum polymers, which we refer to asdirected paths, are identified with isometric embeddings of the unit interval$[0,1]$ into a compact diamond fractal with Hausdorff dimension two, and thereis a natural 'uniform' probability measure, $\mu$, over the space of directedpaths, $\Gamma$. Realizations of the random path measures $\mathbf{M}_{r}$exhibit strong localization properties in comparison to their subcriticalcounterparts when the diamond fractal has dimension less than two. Whereas twopaths $p,q\in \Gamma$ sampled independently using the pure measure $\mu$ haveonly finitely many intersections with probability one, a realization of thedisordered product measure $ \mathbf{M}_{r}\times \mathbf{M}_{r}$ a.s. assignspositive weight to the set of pairs of paths $(p,q)$ whose intersection setsare uncountable but of Hausdorff dimension zero. We give a more refinedcharacterization of the size of these dimension-zero sets using generalized(logarithmic) Hausdorff measures. The law of the random measure$\mathbf{M}_{r}$ cannot be constructed as a subcritical Gaussian multiplicativechaos because the coupling strength to the Gaussian field would, in a formalsense, have to be infinite.

Continuum models of directed polymers on disordered diamond fractals in the critical case