Localization of directed polymers with general reference walk

TL;DR

This paper extends the understanding of localization phenomena in directed polymers to more general reference walks beyond simple random walks, demonstrating atomic endpoint distributions and geometric localization under broad conditions.

Contribution

It proves that low-temperature localization properties hold for any reference walk, generalizing previous results limited to simple or long-range stable processes.

Findings

Polymer endpoint distribution is asymptotically purely atomic.

Geometric localization occurs along a positive density subsequence.

A variational formula for free energy is derived for general reference walks.

Abstract

Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an $\alpha$-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer's endpoint distribution is asymptotically purely atomic, thus strengthening the best known result for long-range directed polymers. A second result proving geometric localization along a positive density subsequence is new to the general case. The proofs use a generalization of the approach introduced by the author with S. Chatterjee in a recent manuscript on the quenched endpoint distribution; this generalization allows one to weaken assumptions on the both the walk and the environment. The methods of this paper also give rise to a variational formula for free energy which is analogous to the one obtained in the simple random walk case.