Operator level limit of the circular Jacobi $\beta$-ensemble

TL;DR

This paper establishes an operator-level limit for the circular Jacobi β-ensemble, characterizing the limiting point process and the convergence of normalized characteristic polynomials to a random analytic function.

Contribution

It introduces a new operator-level limit for the circular Jacobi β-ensemble and characterizes the limit processes and functions via stochastic differential equations.

Findings

Counting function of the limit point process characterized.

Normalized characteristic polynomials converge to a random analytic function.

Results extended to the real orthogonal β-ensemble.

Abstract

We prove an operator level limit for the circular Jacobi $\beta$-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal $\beta$-ensemble.