双曲行列式过程的正则解析实现
Canonical analytic realizations of hyperbolic determinantal processes
- National Yang Ming Chiao Tung University（国立阳明交通大学）
- South China University of Technology（华南理工大学）
- Soochow University（苏州大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造了圆盘上双曲行列式点过程在任意正参数下的随机解析零点实现，证明其 Möbius 协变性并分类了所有此类实现，且与高斯幂级数行列式在整数参数处一致。
AI中文摘要：
Krishnapur 提出一个问题：在非整数参数下，圆盘上的不变双曲行列式点过程是否允许随机解析零点集解释。我们为每个正实数参数构造了这样的实现，作为归一化有限 Blaschke 乘积在分布意义下的完全紧开拓扑极限。零点决定了模和归一化解析形状，留下一个独立的均匀相位。我们证明了精确的 Möbius 协变性，并分类了所有具有此协变性且在原点具有平方可积对数模的实现：它们恰好是正则函数的独立正随机倍数。在此协变类中，匹配正则对数均值和方差唯一确定了正则函数分布。该族在参数上弱连续，并在正整数处与矩阵值高斯幂级数的行列式一致。一个显式的 Barnes $G$ 函数 Mellin 变换确定了基点归一化。
英文摘要：
Krishnapur asked whether the invariant hyperbolic determinantal point processes on the disk admit a random analytic zero-set interpretation at noninteger parameters. We construct such a realization for every positive real parameter as the full compact-open limit in distribution of normalized finite Blaschke products. The zeros determine the modulus and normalized analytic shape, leaving one independent uniform phase. We prove exact Möbius covariance and classify all realizations with this covariance and square-integrable logarithmic modulus at the origin: they are precisely independent positive random multiples of the canonical function. Within this covariant class, matching the canonical logarithmic mean and variance uniquely determines the canonical function law. The family is weakly continuous in the parameter and agrees at positive integers with determinants of matrix-valued Gaussian power series. An explicit Barnes $G$-function Mellin transform determines the basepoint normalization.
