Speed of convergence in the Central Limit Theorem for the determinantal point process with the Bessel kernel

TL;DR

This work derives an exact asymptotic expansion for the Fredholm determinant of Bessel-kernel operators restricted to $[0,R]$, expressing $\det(I + \chi_{[0,R]} B_{e^b} \chi_{[0,R]})$ as $\exp(R c_1^{\mathcal{B}}(b) + c_2^{\mathcal{B}}(b) + c_3^{\mathcal{B}}(b)) Q_R^{\mathcal{B}}(b)$ with an explicitly controlled remainder $Q_R^{\mathcal{B}}(b)$. The analysis hinges on Wiener–Hopf factorization in the Sobolev space $H_1(\mathbb{R})$, a determinant-analogue of Jacobi–Dodgson, and the Basor–Ehrhardt asymptotics for Bessel operators, tying the determinant to concrete constants $c_i^{\mathcal{B}}(b)$. Applying these results to the determinantal point process with the Bessel kernel yields precise speeds of convergence for additive functionals in the Kolmogorov–Smirnov metric, including Gaussian-limit conditions when $c_3^{\mathcal{B}}(b)=\tfrac{1}{2}$. The framework also recovers known special cases for $\nu=\pm\tfrac{1}{2}$ and extends previous sine-process results to an infinite configuration setting.

Abstract

We consider a family of linear operators, diagonalized by the Hankel transform. The Fredholm determinants of these operators, restricted to $L_2[0, R]$, are expressed in a convenient form for asymptotic analysis as $R\to\infty$. The result is an identity, in which the determinant is equal to the leading asymptotic multiplied by an asymptotically small factor, for which an explicit formula is derived. We apply the result to the determinantal point process with the Bessel kernel, calculating the speed of the convergence of additive functionals with respect to the Kolmogorov-Smirnov metric.

Theorems & Definitions (43)

• Theorem 2.1
• Remark
• Theorem 2.2
• Theorem 3.1
• Remark
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Definition 1
• Theorem 4.1: Simon2015