Strong disorder in semidirected random polymers

TL;DR

This paper investigates the behavior of a random walk in a random potential, focusing on the differences between annealed and quenched costs and demonstrating localization phenomena in certain conditions.

Contribution

It identifies conditions where annealed and quenched Lyapounov norms differ and proves localization of the polymer path in these cases.

Findings

Annealed and quenched Lyapounov norms can differ.

Localization occurs in certain disorder regimes.

Conditions for strong disorder are characterized.

Abstract

We consider a random walk in a random potential, which models a situation of a random polymer and we study the annealed and quenched costs to perform long crossings from a point to a hyperplane. These costs are measured by the so called Lyapounov norms. We identify situations where the point-to-hyperplane annealed and quenched Lyapounov norms are different. We also prove that in these cases the polymer path exhibits localization.