Central limit theorem for the determinantal point process with the confluent hypergeometric kernel

TL;DR

The paper analyzes additive functionals of a determinantal point process governed by the confluent hypergeometric kernel. It derives an exact Fredholm-determinant expression for the Laplace transform of regularized sums, isolating a universal Gaussian component and a remainder $Q(f)$ that is continuous and close to 1, enabling a CLT-type Gaussian limit under dilation. The main methodological contribution is the Wiener–Hopf factorization of the associated operators and the use of Ehrhardt’s Helton–Howe framework, together with Jacobi–Dodgson identities, to obtain an explicit determinant formula for $Q(f)$ and control the convergence rate in the Kolmogorov–Smirnov metric. The results generalize known sine-process CLTs ($s=0$) to the confluent hypergeometric setting, connect additive functionals to Wiener–Hopf determinants, and provide quantitative KS bounds, with potential implications for scaling limits and Palm-measure analyses in related determinantal ensembles.

Abstract

We consider the convergence of additive functionals under the determinantal point process with the confluent hypergeometric kernel, corresponding to a sufficiently smooth function $f(x/R)$, as $R\to\infty$. We show that these functionals approach Gaussian distribution and give an estimate on the Kolmogorov-Smirnov distance. To obtain these results we derive an exact identity for expectations of multiplicative functionals in terms of Fredholm determinants.

Theorems & Definitions (51)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1: G_25
• Remark