Localization of directed polymers with general reference walk

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 α-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.2010 Mathematics Subject Classification. 60K37, 82B26, 82B44, 82D60.

Random polymers on the complete graph

“…Remark 1.4. The law m N = m N,1 is the long-time limit of the endpoint distribution, which appears as a fixed point of a transfer operator on Z d for the nearest neighbor graph in [4], and for long range graphs in [3]. (Of course, for a finite state space, there is no question of localization and the disorder is always strong.)…”

“…We make a comment on localization: Directed polymers on the lattice in strong disorder regime have macroscopic atoms, in the sense that the favorite sites at the ending time have mass bounded away from 0 [17,4,3]. Here, the largest mass vanishes at order (log N) −1/α .…”

Localization at the boundary for conditioned random walks in random environment in dimensions two and higher

“…One of the novelties in [3] is a compactification argument for distributions on Z d , which was inspired by the analog treatment for distributions on R d made in [17]. Further applications of similar ideas can be found in [2], [7], and [1].…”

On end-point distribution for directed polymers and related problems for randomly forced Burgers equation

“…In particular, in strong disorder the minimizers exhibit certain localization features, which can be used to understand the limiting behaviour of endpoint distributions. The procedure was extended to other polymer models in [8,9]. We will discuss various localization properties in more detail in the next section.…”