Spherical integrals in probability and beyond

Colin McSwiggen (Academia Sinica)

This mostly expository talk will introduce a number of types of integrals over compact Lie groups that show up constantly in probability and mathematical physics. The analysis of these so-called spherical integrals, in particular their high-dimensional ("large-N") asymptotics, has played a central role in random matrix theory and related subjects over the last 25 years. I'll outline the history of these special functions and describe a number of applications in which they arise, including some original research results with various coauthors. Finally, as a segue to the second lecture by Jon Novak, I'll discuss approaches to large-N analysis.

Hypergeometric functions of huge (random) matrices

Jonathan Novak (UC San Diego)

Hypergeometric functions of matrix arguments are multivariate generalizations of classical hypergeometric functions, and approximating hypergeometric functions of huge matrices is one of the most exciting open problems in high-dimensional analysis. Over the course of the past decade, it has gradually become clear that hypergeometric functions of matrices are discrete analogues of random matrix partition functions. This analogy is clearest for complex matrices, where hypergeometric functions are discrete counterparts of the partition function of the Hermitian one-matrix model with an arbitrary potential, with Schur measure taking on the role of the Gaussian background. Once this is understood, a striking conjecture explicitly describing the asymptotics of all hypergeometric functions emerges. I will explain this conjecture, and outline recent progress towards its solution.