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2018 Localization of directed polymers with general reference walk
Erik Bates
Electron. J. Probab. 23: 1-45 (2018). DOI: 10.1214/18-EJP158


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.


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Erik Bates. "Localization of directed polymers with general reference walk." Electron. J. Probab. 23 1 - 45, 2018.


Received: 26 August 2017; Accepted: 20 March 2018; Published: 2018
First available in Project Euclid: 30 March 2018

zbMATH: 1390.60359
MathSciNet: MR3785400
Digital Object Identifier: 10.1214/18-EJP158

Primary: 60K37
Secondary: 82B26 , 82B44 , 82D60

Keywords: Directed polymers , Free energy , Localization , long-range polymer , phase transition

Vol.23 • 2018
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