Many growth models exhibiting a global smoothing in presence of local roughening are predicted to behave in the same way as a canonical non-linear stochastic partial differential equation known as the Kardar-Parisi-Zhang (KPZ) equation. Although some remarkable bijections to algebraic objects such as random matrices, Young diagrams and so on have led to a series of breakthroughs in the mathematical verification of some of the predictions, many of the current techniques based on integrable probability are not sufficient by themselves to analyze some of the fundamental geometric properties of such systems. Continuing an ongoing program to couple the integrable approach with a primarily probabilistic and geometric perspective, the PI lays down a comprehensive plan to investigate several aspects of such models which will fundamentally improve our understanding and initiate new research directions. The program also has a strong education component including mentoring graduate students and postdocs, and curriculum development at undergraduate and graduate levels, aiming to create multiple advanced research topics courses, and design the probability content in foundational undergraduate courses. Various educational and dissemination strategies including workshop organizing, writing survey articles, teaching summer courses will be carried out as well.
A canonical model of growth predicted to be in the KPZ universality class is the model of planar Last Passage Percolation (LPP) which puts random weights on the vertices of a planar lattice and considers paths between vertices which accrue maximum weights. Such maximal paths called geodesics are fundamental objects of study. Besides further developing the picture of coalescence of geodesics in LPP models, the project aims to study the entire energy landscape of paths and their associated weights, with a focus on the geometry of almost maximal paths or near geodesics. The PI will also investigate interlacing properties of geodesic watermelons (collections of disjoint paths with maximal cumulative weight) and their consequences with connections to various line ensembles and determinantal point processes, as well as large deviation behaviors of geodesics. Finally the program will initiate novel research directions concerning fractal geometry and noise sensitivity, drawing inspiration from seminal works of a similar nature in the context of critical planar percolation and spin glasses. Particular topics include studying Hausdorff dimensions of various endpoint pairs admitting exceptional geodesic behavior as well as computing exponents marking the onset of chaos in natural dynamical versions of LPP. As necessary tools, several new theories including discrete harmonic analysis for models in the KPZ universality class will be developed. In particular, this is expected to create a bridge between various communities in probability, mathematical physics and theoretical computer science.
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.
