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

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 disorder. The continuum polymers are identified with isometric embeddings of the unit interval $[0,1]$ into a compact diamond fractal with Hausdorff dimension two, and there is a natural probability measure, $mu$, identifiable as being `uniform over the space of continuum polymers, $Gamma$. Realizations of the random measures $mathbf{M}_{r}$ exhibit strong localization properties in comparison to their subcritical counterparts when the diamond fractal has dimension less than two. Whereas two directed paths $p,qin Gamma$ chosen independently according to the pure measure $mu$ have only finitely many intersections with probability one, a realization of the disordered product measure $ mathbf{M}_{r}times mathbf{M}_{r}$ a.s. assigns positive weight to the set of pairs of paths $(p,q)$ whose intersection sets are uncountable but with Hausdorff dimension zero. We give a more refined characterization 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 multiplicative chaos because the coupling strength to the Gaussian field would, in a formal sense, have to be infinite.