Zeros of the i.i.d. Gaussian Laurent Series on an Annulus: Weighted Szegő Kernels and Permanental-Determinantal Point Processes

Autor: Makoto Katori, Tomoyuki Shirai
Rok vydání: 2022
Předmět:
Zdroj: Communications in Mathematical Physics. 392:1099-1151
ISSN: 1432-0916
0010-3616
DOI: 10.1007/s00220-022-04365-2
Popis: 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 Szeg�� 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 Szeg�� 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 Vir��g ($r \to 0$) and that DPP of Peres and Vir��g with a deterministic zero added at the origin ($r \to \infty$).
v3: LaTeX 58 pages, 3 figures
Databáze: OpenAIRE
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