Systems with many degrees of freedom: probalistic and constructive field theory methods

在我们第一个资助期的进展的基础上，我们将考虑某些类型的具有多个自由度的复杂系统，并调查它们行为的新的基本方面。对于玻色子多粒子系统，我们希望找到固定温度下玻色-爱因斯坦凝聚的数学证据，并研究在消失温度下正距离粒子的附加引力的影响。对于费米子多粒子系统，我们希望将我们在第一个资助期完成的分析扩展到二维系统和在较弱稀释性假设下的三维系统。在抛物型Anderson模型中，我们希望研究系统的老化特性以及时间上的关联，以期研究一般的随机跳跃动力学。同样，所有这些任务都需要系统地结合概率和分析工具。

Building on our progress of the first funding period, we will consider certain types of complex systems with many degrees of freedom and investigate new fundamental aspects of their behaviour. For bosonic many-particle systems, we want to find mathematical evidence for Bose-Einstein condensation at fixed temperature, and we want to study the effect of an additional attraction of the particles at positive distance at vanishing temperature. For fermionic many-particle systems, we want to extend the analysis we achieved in the first funding period to two-dimensional systems and to three-dimensional systems under a weaker diluteness assumption. In the parabolic Anderson model we want to study aging properties of the system and correlations in time, in the hope to investigate random hopping dynamics in general. Again, all these tasks require a systematic combination of probabilistic and analytic tools.