Path properties of the disordered pinning model in the delocalized regime

TL;DR

This paper investigates the behavior of a disordered polymer model in the delocalized regime, showing that contacts with the defect line are probabilistically tight, but can grow logarithmically at low temperatures along subsequences.

Contribution

It provides new insights into the path properties of disordered polymers, particularly characterizing contact number behavior in the delocalized phase across temperatures.

Findings

Number of contacts remains tight in probability at any temperature.

At low temperature, contacts grow logarithmically along subsequences.

The results hold for the entire delocalized regime.

Abstract

We study the path properties of a random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense "tight in probability" as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.