## Electronic Journal of Probability

### Localization of directed polymers with general reference walk

Erik Bates

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 30, 45 pp.

Dates
Accepted: 20 March 2018
First available in Project Euclid: 30 March 2018

https://projecteuclid.org/euclid.ejp/1522375270

Digital Object Identifier
doi:10.1214/18-EJP158

Mathematical Reviews number (MathSciNet)
MR3785400

Zentralblatt MATH identifier
1390.60359

#### Citation

Bates, Erik. Localization of directed polymers with general reference walk. Electron. J. Probab. 23 (2018), paper no. 30, 45 pp. doi:10.1214/18-EJP158. https://projecteuclid.org/euclid.ejp/1522375270

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