Random Matrices, Complexity and Slow Dynamics in Random Media

无序和随机性会导致系统的深度复杂性。本项目研究了这种诱导复杂性对此类系统时间演化的重要性。因此，它解决了在无序介质的数学和统计物理之间的接口的主题。在这些复杂系统的动力学上，复杂性的特征体现在这样一个事实中：进化非常缓慢，并且被困在远离平衡的区域。它表现出老化等新特征，打破了已有的涨落耗散规律。然而，这些复杂的系统有一些共同的特征，这个项目为理解这些普遍现象奠定了基础。这种普遍的行为表现在非常不同的模型中，包括自旋玻璃的慢动力学，包括球形和伊辛自旋，以及像渗透团簇和随机树这样的无序结构上的扩散和输运。最近（由PI）建立的工具之一是在随机能量景观和随机矩阵理论的研究之间建立了一个新的强有力的联系。这种联系还将被进一步探索，因为它可以在一类系统（球形自旋玻璃）中引入不同类别的复杂行为的新分类。

Disorder and randomness can induce very deep complexity for systems. This project studies the importance of this induced complexity on the time evolution of such systems. It thus addresses topics at the interface between mathematics and statistical physics of disordered media. The signature of complexity on the dynamics of these complex systems is seen in the fact that the evolution is very slow and trapped in regions far from equilibrium. It exhibits new characteristics like aging, and break the well established rules of fluctuation-dissipation. Nevertheless some universal features are shared by these complex systems and this project builds the foundations for understanding these universal phenomena. This universal behavior is exhibited in very different models which include the slow dynamics of Spin glasses, both for spherical and Ising spins, and the diffusion and transport on disordered structures like percolation clusters and random trees. One of the tools recently built (by the PI) is a new and powerful link between the study of the random energy landscapes and random matrix theory. This link will also be explored further since it enables the introduction of a new classification of different classes of complex behavior in one class of systems (the spherical spin glasses).