Localization of directed polymers in continuous space
Localization of directed polymers in continuous space
复制标题
DOI：
10.1214/20-ejp530
复制
发表时间：
2019-05
影响因子：
1.4
通讯作者：
Yuri Bakhtin;Donghyun Seo
中科院分区：
文献类型：
--
作者：
Yuri Bakhtin;Donghyun Seo
本文的第一个主要目的是给出Mukherjee-Varadhan拓扑的一种新的度量化，它是最近引入的欧氏空间上概率测度空间的平移不变紧化。这一新的度量化使我们能够实现我们的第二个目标，即扩展Bates和Chatterjee最近关于基于欧氏空间中一般随机游动的离散定向聚合物到聚合物的端点分布的局部化程序。遵循他们的策略，我们研究了端点分布更新映射的渐近行为，并研究了它的分布不动点集满足变分原理。我们证明了当且仅当系统处于高温区时，集中于零度量点的分布是该集合中唯一的元素。这使得我们能够证明渐近团簇(渐近纯原子性性质的自然连续模拟)在低温区域成立，并且当且仅当系统处于低温区域时，端点分布以正密度几何定域。
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 distributdion 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.