# Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes

### Abstract

The infinite-dimensional unitary group ${\rm U}(\infty)$ is the inductive limit of growing compact unitary groups ${\rm U}(N)$. In this paper we solve a problem of harmonic analysis on ${\rm U}(\infty)$ stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable $4$-parameter family of characters of ${\rm U}(\infty)$. These characters generate representations which should be viewed as analogs of nonexisting regular representation of ${\rm U}(\infty)$.

The spectral decomposition of a character of ${\rm U}(\infty)$ is described by the spectral measure which lives on an infinite-dimensional space $\Omega$ of indecomposable characters. The key idea which allows us to solve the problem is to embed $\Omega$ into the space of point configurations on the real line without two points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special ‘integrable’ form and are expressed through the Gauss hypergeometric function.

From the analytic point of view, the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by Richard Askey and Peter Lesky. One difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to $0$. We do this by expressing the polynomials in terms of a solution of a discrete Riemann-Hilbert problem and computing the (nonoscillatory) asymptotics of this solution.

From the point of view of statistical physics, we study thermodynamic limit of a discrete log-gas system. An interesting feature of this log-gas is that its density function is asymptotically equal to the characteristic function of an interval. Our point processes describe how different the random particle configuration is from the typical ‘densely packed’ configuration.

In simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices, similar results were obtained in our papers [BO1], [BO2], [BO4].

## Authors

Alexei Borodin

Department of Mathematics
California Institute of Technology