Slow Dynamics in Random Media

New York University, New York NY

Investigators

Abstract

This project addresses topics at the interface between probability theory and statistical physics of disordered media. It proposes to build a systematic understanding of the slowing down or trapping of stochastic dynamics of disordered systems. The understanding is that for some of these systems the slowing down is due to localized regions in configuration space where the system spends most of its time. A general paradigm (the Trap Model) has been introduced in physics literature and studied recently by the PI and various collaborators. This project proposes to do three things: sharpen this understanding of this phenomenological model, show that it is relevant for mean-field Spin Glass dynamics (and aging) in a wide range of time scale (building on recent work of the PI and collaborators), and introduce the point of view of trap models in hard problems of Random Walks on percolation clusters. The PI will investigate the first topic and second topic with graduate students and an established international network of collaborators. The Co-PI will investigate one of the two themes of the third topic (Biased random walks on supercritical clusters) with the PI for the duration of his appointment at Courant Institute (approximately two years). The second topic will be investigated by the PI, the Co-PI and external collaborators. Many branches of modern science are challenged with the complicated task of attaining a clear understanding of the time evolution of complex systems, or more specifically, of large systems comprised of many components and with complex local rules. In particular, the lessons learned in the last century in regards to the properties of statistical equilibrium seem of little value for most of these complex systems as they tend to be, under usual conditions, always very far from equilibrium and, in a sense, always in a transient regime. They achieve their equilibrium very slowly and exhibit fascinating properties along the way; one of them being "aging". These very slow evolutions are the subject of this proposal for addressing well chosen paradigms which are known to be complex and "universal", i.e. representative of a wide class of such phenomena.