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

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.