Localization of directed polymers in continuous space

Yuri Bakhtin, Donghyun Seo

Published 2019-05-02Version 1

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 prove that it has a unique fixed point satisfying a variational principle if and only if the system is in the low temperature regime. This enables us to prove that the the asymptotic clusterization (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.