Exactly Solvable Stochastic Systems: Connections and Universality

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This project deals with large random systems, which can be typically described as collections of particles with certain prescribed rules of interactions. Examples are given by random stepped surfaces in three-dimensional space, random sorting networks, six-vertex model (also known as the square ice and used to model a thin planar layer of molecules of water). A distinguishing feature of the studied systems is the presence of numerous exact formulas, describing the probabilistic characteristics of the systems. The central questions concern asymptotic properties of the systems of growing sizes, and the research aims at exact rather than qualitative answers.The project aims at two kinds of results. First, the new types of asymptotic behavior lead to discoveries of connections between stochastic systems of different origins, such as lattice models of two-dimensional statistical mechanics, random matrices, stochastic partial differential equations, and probability measures of asymptotic representation theory. Second, the development of robust methods leads to the extensions of the results from the exactly solvable cases to much wider universality classes, thus, justifying the use of these cases for the extensive peculiar predictions, potentially reaching the real-world applications. In particular, during the project the PI will investigate the transition between Gaussian Free Field and Gaussian fields solving hyperbolic SPDEs in the six-vertex model, connections between random sorting networks and eigenvalue distributions of random matrices, and the link between matrix elements of powers of random Hermitian matrices and the finite-time distribution of the Kardar-Parisi-Zhang SPDE.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.