Joint Continuity and Selberg--ODE Equivalence for the Sineβ Pair Correlation Function
Abstract
We address two questions posed by Qu and Valkó in their study of the pair correlation function of the Sineβ process. First, we prove that the pair correlation function admits a jointly continuous version in the inverse-temperature and spatial parameters on (0,∞)×ℝ, removing the restriction β>2 in their joint-continuity result. The argument uses a general observation: separate weak continuity of a family of probability laws, together with stochastic monotonicity in one parameter, implies joint weak continuity. Applied to the terminal value of the Qu--Valkó diffusion, their Palm-density formula then gives the result without differentiating the Fourier expansion. Second, for β=2n we give a direct proof that the Qu--Valkó matrix-recursion power series agrees with the Selberg-integral representation recorded by Forrester. A Krawtchouk transform converts the Qu--Valkó system into a (2n+1)-dimensional differential system with parameter −(n+1), while a centered Aomoto--Selberg trace system has parameter n+1. An explicit triangular differential intertwiner reflects the parameter η↦−η. Matching the unique Frobenius branch of exponent $2$ gives exactly the Selberg normalization. As a consequence, every coefficient of the Qu--Valkó recursion is identified with an even centered moment of the corresponding Jacobi--Selberg trace statistic.
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BibTeX
@misc{fang2026joint,
title = {{Joint Continuity and Selberg--ODE Equivalence for the $\mathrmSine_β$ Pair Correlation Function}},
author = {Weiyang Fang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17596},
eprint = {2609.17596},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17596},
}
RIS
TY - UNPB
TI - Joint Continuity and Selberg--ODE Equivalence for the Sine_β Pair Correlation Function
AU - Weiyang Fang
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.17596
UR - https://arxiv.org/abs/2609.17596
AB - We address two questions posed by Qu and Valkó in their study of the pair correlation function of the Sine_β process. First, we prove that the pair correlation function admits a jointly continuous version in the inverse-temperature and spatial parameters on (0,∞)×ℝ, removing the restriction β>2 in their joint-continuity result. The argument uses a general observation: separate weak continuity of a family of probability laws, together with stochastic monotonicity in one parameter, implies joint weak continuity. Applied to the terminal value of the Qu--Valkó diffusion, their Palm-density formula then gives the result without differentiating the Fourier expansion. Second, for β=2n we give a direct proof that the Qu--Valkó matrix-recursion power series agrees with the Selberg-integral representation recorded by Forrester. A Krawtchouk transform converts the Qu--Valkó system into a (2n+1)-dimensional differential system with parameter −(n+1), while a centered Aomoto--Selberg trace system has parameter n+1. An explicit triangular differential intertwiner reflects the parameter η↦−η. Matching the unique Frobenius branch of exponent $2$ gives exactly the Selberg normalization. As a consequence, every coefficient of the Qu--Valkó recursion is identified with an even centered moment of the corresponding Jacobi--Selberg trace statistic.
ER -