Asymptotics and Structure of Random Matrices and Random Partitions by means of Special Functions

A goal of probability theory is to understand and predict the behavior of large random systems that are present in everyday life, such as traffic models, percolation, polymers, and the molecular structure of solids. Because a rigorous treatment of such random systems is still out of the reach of probability theory, mathematicians and physicists instead consider idealized models with the same large-scale behaviors that are amenable to standard mathematical analysis. In this project, the PI will use multivariate special function techniques to construct and study random idealized systems that are “solvable,” meaning they can be analyzed through well-honed, yet sophisticated, ideas from representation theory and combinatorics. At the same time, these models are ubiquitous in nature, because of their origins in random matrices, plane partitions and quantum algebras. This project will provide training activities and outreach efforts aimed at undergraduate and graduate students through research opportunities and lecture series led by the PI. This project is centered around two interrelated topics: (i) Continuous and discrete Boltzmann distributions from statistical physics. These can be regarded as one-parameter generalizations of random matrix eigenvalues and random partitions. The goal is to describe, using novel versions of free probability, the large-scale asymptotics of random particle ensembles at various temperature limit regimes. The main tool will be the Dunkl transform, which depends on multivariate Bessel functions. (ii) Asymptotic representation theory and its relations with beta-ensembles. The open problems to be considered are infinite-dimensional versions of the classification of irreducible representations and the problem of noncommutative harmonic analysis. Prior work employed q-orthogonal multivariate polynomials to unveil links between representations and novel discrete quantized analogues of beta-ensembles, yielding connections with measures from topic (i). 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.