AI 中文总结
本文证明Sine_β配对相关函数在β>0时具有联合连续性，并针对β=2n情形建立了Qu-Valkó递归与Selberg积分表示的等价性，从而识别出所有系数为Jacobi-Selberg迹统计量的偶数中心矩。
AI 中文摘要
我们解决了Qu和Valkó在研究$\mathrm{Sine}_\beta$过程的配对相关函数时提出的两个问题。首先，我们证明了配对相关函数在$(0,\infty)\times\mathbb{R}$上的逆温度和空间参数上具有联合连续版本，去除了他们联合连续性结果中$\beta>2$的限制。该论证使用了一个一般性观察：一族概率律的分别弱连续性，加上在一个参数上的随机单调性，蕴含联合弱连续性。应用于Qu--Valkó扩散的终值，他们的Palm密度公式随后给出结果，而无需对Fourier展开进行微分。其次，对于$\beta=2n$，我们直接证明了Qu--Valkó矩阵递归幂级数与Forrester记录的Selberg积分表示一致。一个Krawtchouk变换将Qu--Valkó系统转化为参数为$-(n+1)$的$(2n+1)$维微分系统，而一个中心化的Aomoto--Selberg迹系统具有参数$n+1$。一个显式的三角微分交织子反映了参数$\eta\mapsto-\eta$。匹配指数为2的唯一Frobenius分支恰好给出Selberg归一化。因此，Qu--Valkó递归的每个系数都被识别为相应Jacobi--Selberg迹统计量的偶数中心矩。
英文摘要
We address two questions posed by Qu and Valkó in their study of the pair correlation function of the $\mathrm{Sine}_β$ process. First, we prove that the pair correlation function admits a jointly continuous version in the inverse-temperature and spatial parameters on $(0,\infty)\times\mathbb{R}$, 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 $η\mapsto-η$. 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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