论文标题
I.I.D.的零Gaussian Laurent系列在环上：加权的Szegő内核和永久性确定点过程
Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes
论文作者
论文摘要
在一个环上$ {\ mathbb {a}} _ q：= \ {z \ in {\ mathbb {c}}}：q <| z | <1 \} $带有固定的$ q \ in（0，1）$，我们研究高斯分析功能（GAF）及其零集，该集合在$ {\ m athbb {a}} _ q $上定义了一个点过程，称为GAF的零点进程。 GAF由I.I.D.〜gaussian Laurent系列定义，因此，由$ r> 0 $参数化的协方差内核与加权的szegő内核，$ {\ mathbb {a} a}} _ q $，以及由McCullough and Shen研究的权重参数$ r $。 GAF和零点过程是旋转不变的，并且具有与坐标$ z \ leftrightArrow q/z $的$ q $ inversion相关的对称性，并且参数更改$ r \ leftrightArrow q^2/r $。当$ r = q $时，它们是在保留$ {\ mathbb {a}} _ q $的共形变换下不变的。通过添加零来调节GAF，诱导了新的GAF，以使协方差内核也由McCullough和Shen的加权Szegő内核给出，但重量参数$ r $根据添加的零更改。 我们还证明，GAF的零点过程提供了一个永久性确定点过程（PDPP），其中每个相关函数由永久性乘以确定性的次数表示。研究了对PDPP展开的2相关函数的$ r $的依赖。如果我们采用限制$ q \ to 0 $，则在单位磁盘$ {\ mathbb {d}} $上获得更简单但仍然非平凡的PDPP。我们观察到，$ r \ in（0，\ infty）$索引的极限PDPP可以被视为$ {\ Mathbb {d}} $在$ {\ Mathbb {d}}上的确定点过程（DPP）之间的插值\ infty $）。
On an annulus ${\mathbb{A}}_q :=\{z \in {\mathbb{C}}: q < |z| < 1\}$ with a fixed $q \in (0, 1)$, we study a Gaussian analytic function (GAF) and its zero set which defines a point process on ${\mathbb{A}}_q$ called the zero point process of the GAF. The GAF is defined by the i.i.d.~Gaussian Laurent series such that the covariance kernel parameterized by $r >0$ is identified with the weighted Szegő kernel of ${\mathbb{A}}_q$ with the weight parameter $r$ studied by Mccullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the $q$-inversion of coordinate $z \leftrightarrow q/z$ and the parameter change $r \leftrightarrow q^2/r$. When $r=q$ they are invariant under conformal transformations which preserve ${\mathbb{A}}_q$. Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of Mccullough and Shen but the weight parameter $r$ is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on $r$ of the unfolded 2-correlation function of the PDPP is studied. If we take the limit $q \to 0$, a simpler but still non-trivial PDPP is obtained on the unit disk ${\mathbb{D}}$. We observe that the limit PDPP indexed by $r \in (0, \infty)$ can be regarded as an interpolation between the determinantal point process (DPP) on ${\mathbb{D}}$ studied by Peres and Virág ($r \to 0$) and that DPP of Peres and Virág with a deterministic zero added at the origin ($r \to \infty$).
