Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes

Abstract

Rosengren and Schlosser introduced notions of ${\it R}_N$-theta functions for the seven types of irreducible reduced affine root systems, ${\it R}_N={\it A}_{N-1}$, ${\it B}_{N}$, ${\it B}^{\vee}_N$, ${\it C}_{N}$, ${\it C}^{\vee}_N$, ${\it BC}_{N}$, ${\it D}_{N}$, $N \in \mathbb{N}$, and gave the Macdonald denominator formulas. We prove that, if the variables of the ${\it R}_N$-theta functions are properly scaled with $N$, they construct seven sets of biorthogonal functions, each of which has a continuous parameter $t \in (0, t_{\ast})$ with given $0< t_{\ast} < \infty$. Following the standard method in random matrix theory, we introduce seven types of one-parameter ($t \in (0, t_{\ast})$) families of determinantal point processes in one dimension, in which the correlation kernels are expressed by the biorthogonal theta functions. We demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions. In the scaling limits associated with $N \to \infty$, we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter $t \in (0, t_{\ast})$. We give new expressions for the Macdonald denominators using the Karlin--McGregor--Lindstr\"om--Gessel--Viennot determinants for noncolliding Brownian paths, and show the realization of the associated elliptic determinantal point processes as noncolliding Brownian brides with a time duration $t_{\ast}$, which are specified by the pinned configurations at time $t=0$ and $t=t_{\ast}$.