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

TL;DR

This paper constructs and analyzes continuum directed polymer measures on disordered diamond fractals, revealing strong localization and complex intersection properties in the critical regime, which differ from subcritical models.

Contribution

It introduces a new family of continuum polymer measures on fractals in the critical case, with detailed intersection and localization properties, extending understanding of disordered systems.

Findings

Realizations exhibit strong localization compared to subcritical cases.

Paths under the disordered measure have uncountably many intersections with Hausdorff dimension zero.

The measure cannot be obtained via subcritical Gaussian multiplicative chaos.

Abstract

We construct and study a family random continuum 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, which we refer to as directed paths, 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 'uniform' probability measure, $\mu$, over the space of directed paths, $\Gamma$. Realizations of the random path 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 paths $p,q\in \Gamma$ sampled independently using 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 of 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.