Large deviation analysis of interacting systems of Brownian motions and random interlacements at positive temperature

总的主题是相互作用的粒子系统及其临界现象。其新奇之处在于将相互作用布朗运动的大偏差分析与最近发展的随机交错理论相结合。第二个新奇之处是对散射长度的特定标度，它描述了布朗运动和随机交错的相互作用。目的是证明无限长循环上的凝聚是相互作用玻色子的所谓玻色爱因斯坦凝聚的信号。这一结果将首次建立概率凝聚及其在数学物理中的应用。主要的新奇之处在于证明了“无限长”的周期表现为某种随机的交错过程。其目的是找到这样一个随机过程。它主要是一个使用大偏差技术、随机分析、多尺度分析和相互作用的粒子系统的概率项目。在这个项目中，分析了最吸引人和最具挑战性的模型之一，即正温度下的相互作用玻色子。自从20世纪90年代末的冷原子实验和两次诺贝尔奖以来，数学研究的目标是证明所谓的玻色-爱因斯坦凝聚临界现象，比如液氦在低温下的超流动性。该项目使用概率方法来研究量子相互作用系统--所谓的Feynman-Kac公式允许将量子问题转化为概率论中的经典问题。主要使用的技术是大偏差分析、随机分析和浓度不等的变体。具体地说，该项目研究了带相互作用的标记泊松点过程，该项目涉及以下步骤：(1)在Gross-Pitaevskii标度极限中，考察了相互作用的布朗运动的大N极限(布朗运动的个数)和相互作用的布朗运动的大时间极限。第一步是发展和使用所谓的散射长度的概率版本，以获得相互作用项的尺度作用的令人信服的描述。(2)第二步是研究具有周期长度分布和交错分布的标记点过程(这两个系统都没有相互作用项)。并将玻色-爱因斯坦凝聚现象与所谓的随机交织上正概率权的出现联系起来，随机交错是双无限(时间视界)路径(来自无穷远并消失到无限远的路径)的随机过程。随机交错是一类新颖的过程，最近获得了大量的研究活动。我们的目标是展示这一新概念在分析类玻色-爱因斯坦凝聚现象中的有用应用。(3)一旦步骤(2)证明了凝聚，主要部分将考虑有或没有随机交错过程的有限圈之间的相互作用。作为前一步，该项目将证明步骤(1)的正散射长度可以引发凝聚现象。一旦项目完成了这一分析，项目就会研究系统，而不需要扩展交互术语。这一步是整个项目中最具挑战性的一步，因为它旨在展示空间相关性和交错凝聚之间的联系。这一结果将在该领域取得突破，并很有可能对多粒子系统及其凝聚现象的方向产生持久的影响。(4)一旦步骤3显示了新的凝聚现象，该项目将研究Gibbs度量在布朗运动和随机交织耦合系统中的作用。

The overall theme is interacting particle systems and their critical phenomena. The novelty is to combine large deviation analysis for interacting Brownian motions with the recently developed theory of random interlacements. A second novelty is the specific scaling towards the scattering length, which describes interactions of Brownian motions and random interlacements. The aim to prove that condensation onto infinitely long cycles, which are given as random interlacements, is a signal of the so-called BoseEinstein condensation for interacting Bosons. This result will for the first time establish probabilistic condensation and its applications to mathematical physics. The main novelty is to prove that 'infinitely long' cycles appear as some random interlacements process. The aim is to find such a random process. It is primarily a probabilistic project using large deviation techniques, stochastic analysis, multi-scale analysis and interacting particle systems.Background. In this project, one of the most fascinating and challenging models is analysed, namely, interacting Bosons at positive temperature. Since the experiments on cold atoms in the late 1990s and two Nobel Prizes, mathematical research has started aiming to prove the so-called Bose-Einstein condensation critical phenomena like for example the superfluidity of liquid Helium at low temperatures. The project is using probabilistic methods for the quantum interacting systems - the so-called Feynman-Kac formula allows to transfer the quantum problem to a classical problem in probability theory.Main techniques to be used are variants of large deviation analysis, stochastic analysis and concentration inequalities. In particular, the project studies marked Poisson point processes with interaction, and the project involves the following steps:(1) Examination of the large N limit (number of Brownian motions) coupled with the large time limit for interacting Brownian motions in trap potential in the Gross-Pitaevskii scaling limit. The first step is to develop and employ a probabilistic version of the so-called scattering length to obtain a compelling description of the role of scaling of interaction terms.(2) The second step is to study the marked point process with cycle length distributions and interlacements distributions (both systems with no interaction terms). And relate the Bose-Einstein condensation phenomenon to the onset of positive probability weight on the so-called random interlacements, a random process of double-infinite (time horizon) paths (paths coming from infinity and disappearing to infinity).Random interlacements are a novel class of process and have recently attained a lot of research activity. We aim to showcase a useful application of this novel notion for the analysis of Bose-Einstein condensation-like phenomena.(3) Once step (2) proves the condensation, the major part will be to allow for interactions among finite cycles with or without the random interlacement processes. As a previous step, the project will show that the positive scattering length of step (1) can trigger condensation phenomena. Once the project finishes this analysis, the project studies the systems without the scaling of the interaction terms. The step is the most challenging of the whole project as it aims to demonstrate connections between spatial correlations and condensation onto interlacements. This result will establish a breakthrough in the field and is quite likely to have a lasting impact in the direction of many-particle systems and their condensation phenomena.(4) Once step 3 shows the novel condensation phenomena, the project will study the role of Gibbs measures for the coupled systems of Brownian motions and random interlacements.