Open Access
2020 Localization of directed polymers in continuous space
Yuri Bakhtin, Donghyun Seo
Electron. J. Probab. 25: 1-56 (2020). DOI: 10.1214/20-EJP530


The first main goal of this article is to give a new metrization of the Mukherjee–Varadhan topology, recently introduced as a translation-invariant compactification of the space of probability measures on Euclidean spaces. This new metrization allows us to achieve our second goal which is to extend the recent program of Bates and Chatterjee on localization for the endpoint distribution of discrete directed polymers to polymers based on general random walks in Euclidean spaces. Following their strategy, we study the asymptotic behavior of the endpoint distribution update map and study the set of its distributional fixed points satisfying a variational principle. We show that the distribution concentrated on the zero measure is a unique element in this set if and only if the system is in the high temperature regime. This enables us to prove that the asymptotic clustering (a natural continuous analogue of the asymptotic pure atomicity property) holds in the low temperature regime and that the endpoint distribution is geometrically localized with positive density if and only if the system is in the low temperature regime.


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Yuri Bakhtin. Donghyun Seo. "Localization of directed polymers in continuous space." Electron. J. Probab. 25 1 - 56, 2020.


Received: 2 May 2019; Accepted: 3 October 2020; Published: 2020
First available in Project Euclid: 12 December 2020

MathSciNet: MR4186261
Digital Object Identifier: 10.1214/20-EJP530

Primary: 60E05 , 60K37 , 82B26 , 82B44 , 82D60

Keywords: Directed polymers , Localization , Mukherjee-Varadhan topology , phase transition

Vol.25 • 2020
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