Landscapes, games and permutations: Scaled random dynamics in statistical mechanics

The essence of a many-body random system is often captured by its scaling limit, the structure arising when the system’s characteristic scales of space and time are brought to unit order by a linear transformation. And it is often profitable to introduce a dynamical enhancement of the system, whose snapshot at any given moment of time is the system itself, because the enhancement may have attractive properties that the original lacks, or because the behavior of the enhanced system at exceptional moments reflects aspects of the system’s scaling limit. The proposal addresses several random scaling limits and random dynamic enhancements of statistical mechanical systems. This project involves graduate student training. Principal directions of inquiry include study of the random scaled dynamics of a random walker whose attempted linear progress is frustrated by hard obstacles that clutter the route forward; the scale of time in a randomly evolving energy landscape that witnesses convergence to equilibrium of the depth of the deepest valley; and the scaled counter trajectory in random-turn games played on domains in Euclidean space. The proposal will develop and exploit basic tools from the theory of Ornstein-Zernike decay and subcritical connections, and discrete harmonic analysis. By dissemination, mentorship and collaboration, the PI will seek to ensure that the research enhances the mathematical experience and trajectory of junior researchers including graduate students via joint research to develop fundamental tools, and undergraduates via research study and coding projects. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.