Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szeg\H{o} kernels and permanental-determinantal point processes

On an annulus ${\mathbb{A}}_q:=\{z\in \mathbb{C}:q<\mathrm{\mid }z\mathrm{\mid }<1\}$ 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\H{o} 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 $z\leftrightarrow q\mathrm{/}z$ and the parameter change $r\leftrightarrow q^2\mathrm{/}r$. 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\H{o} 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 $r\in (0,\mathrm{\infty })$ can be regarded as an interpolation between the determinantal point process (DPP) on $\mathbb{D}$ studied by Peres and Vir'ag ($r\to 0$) and that DPP of Peres and Vir'ag with a deterministic zero added at the origin ($r\to \mathrm{\infty }$).