Large Scale Asymptotics of Random Spatial Processes: Scaling Exponents, Limit Shapes, and Phase Transitions

点击翻译按钮获取中文摘要

Many natural processes such as growth of bacteria, fluid spreading in a porous medium, directed polymers in random media, propagation of flame fronts and so on, are believed to exhibit various universal properties if observed at certain characteristic spatial and time scales. Much of the research in probability and statistical physics involves investigating random structures equipped with spatial geometry, expected to model some natural phenomena as above and others. The research program outlined in the project aims to study a wide range of problems around various aspects of such random spatial models including correlation structure, scaling limits, phase transitions as certain natural parameters are varied, convergence to equilibrium as well as behavior in the large deviation regimes. While the main focus is on developing novel ideas in probability theory, a key goal is to merge perspectives and develop new bridges between various areas of mathematics, statistical physics and theoretical computer science. The program also has a significant education component including curriculum development at undergraduate and graduate levels, and mentoring graduate students and postdocs. The project broadly discusses three topics. The first theme includes models of random growth exhibiting a global smoothing mechanism in presence of local roughening forces believed to exhibit certain universal behavior predicted in a seminal paper by Kardar, Parisi and Zhang (KPZ). The PI will study models of planar last and first passage percolation, which puts random weights on the vertices of a planar lattice and considers paths between vertices which accrue maximum or minimum energies respectively, and are believed to be canonical examples in the KPZ universality class. There has been an explosion of activity, mostly around a handful of examples of such models, which are integrable, admitting certain remarkable bijections to algebraic objects such as random matrices, Young diagrams and so on. The PI will pursue a geometric perspective and develop probabilistic tools to study spatial and temporal correlation behavior for such models as well as how the geometry of optimal paths change in large deviation regimes. The second theme concerns models of self organized criticality where systems under their natural evolution converge to a critical state without external tuning of parameters. Continuing previous work, the PI will investigate long standing conjectures about phase transitions on infinite lattices and quantitative estimates for finite versions, for the stochastic sandpile model and activated random walk, two paradigm examples of self-organized criticality. The study of evolving self-similar interfaces of related multi-type Laplacian growth models where growth rate is governed by harmonic measure of random walk is also proposed. The final topic is about the study of exponents related to rate of escape, spectral behavior and convergence to equilibrium for random walks and finite Markov chains. Examples considered include models of particles diffusing under gravity in a random evolving potential with connections to fluid mechanics, random walks on random fractal graphs as well as a class of non-monotone spin systems modeling the 'cage effect' in glassy dynamics, with connections to random walk on matrices and oriented percolation.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.