Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes

Technical Summary: Zeros of the i.i.d. Gaussian Laurent Series on an Annulus

Problem and Context
The paper investigates the zero point processes of Gaussian Analytic Functions (GAFs) defined on an annulus $A_q := \{z \in \mathbb{C} : q < |z| < 1\}$ with a fixed $q \in (0, 1)$. This work extends the foundational results of Peres and Virág [64], who studied the zeros of a GAF on the unit disk $\mathbb{D}$ defined by an i.i.d. Gaussian power series. In the disk case, the covariance kernel is the Szegő kernel $S_{\mathbb{D}}(z, w)$, and the resulting zero point process is a Determinantal Point Process (DPP) with correlation kernel given by the Bergman kernel $K_{\mathbb{D}}(z, w) = S_{\mathbb{D}}(z, w)^2$.

The authors aim to generalize this framework to the multiply connected domain $A_q$. They define a GAF $X^r_{A_q}$ via an i.i.d. Gaussian Laurent series whose covariance kernel is identified with a weighted Szegő kernel $S_{A_q}(z, w; r)$, parameterized by a weight $r > 0$. The central problem is to characterize the statistical properties of the zero set of this GAF, specifically its correlation functions, symmetries, and behavior under conditioning.

Methodology
The study relies on the theory of Reproducing Kernel Hilbert Spaces (RKHS) and special functions, particularly theta functions and Weierstrass elliptic functions.

1. Weighted Szegő Kernel: The authors utilize the weighted Szegő kernel $S_{A_q}(z, w; r)$ for the Hilbert space $H^2_r(A_q)$, as studied by Mccullough and Shen [56]. This kernel is expressed using the Jordan–Kronecker function and theta functions:
   $$S_{A_q}(z, w; r) = \frac{q_0^2 \theta(-rwz)}{\theta(-r, wz)}$$
   where $\theta(\cdot)$ is the theta function with nome $p=q^2$.
2. Conditional Kernels: A key methodological step involves analyzing the conditional GAF given that zeros are fixed at specific points $\alpha_1, \dots, \alpha_n$. The authors employ the Mccullough-Shen formula, which states that the conditional kernel retains the form of a weighted Szegő kernel but with a modified weight parameter $r' = r \prod_{\ell=1}^n |\alpha_\ell|^2$.
3. Permanental-Determinantal Structure: The correlation functions of the zero point process are derived using a general formula for GAF zeros [64, 71]. The authors introduce the concept of a "perdet" (permanent multiplied by determinant) to express these correlations. They leverage identities involving theta functions, such as the elliptic extension of Cauchy’s determinant evaluation due to Frobenius, to simplify the expressions.
4. Asymptotic Analysis: To analyze the interaction between zeros, the authors study the unfolded 2-correlation function $g_{A_q}(z, w; r)$. They utilize Weierstrass $\wp$-functions and their properties to determine the asymptotic behavior of correlations at short and long distances.

Key Contributions and Results

1. Permanental-Determinantal Point Process (PDPP):
   The primary result (Theorem 1.3) is that the zero point process $Z_{X^r_{A_q}}$ is a Permanental-Determinantal Point Process (PDPP). The $n$-point correlation function $\rho^n_{A_q}$ is given by:
   $$\rho^n_{A_q}(z_1, \dots, z_n; r) = \frac{\theta(-r)}{\theta(-r \prod_{k=1}^n |z_k|^4)} \text{perdet}_{1 \le i,j \le n} \left[ S_{A_q}\left(z_i, z_j; r \prod_{\ell=1}^n |z_\ell|^2\right) \right]$$
   This structure indicates that the correlation kernel depends hierarchically on the positions of the points being measured, via the weight parameter.

2. Symmetries and Conformal Invariance:

   • (q, r)-Inversion Symmetry: The GAF and its zero process exhibit a symmetry under the transformation $z \mapsto q/z$ combined with a parameter change $r \mapsto q^2/r$ (Proposition 1.1).
   • Conformal Invariance: When the weight parameter is set to $r=q$, the GAF $X_{A_q}$ and its zero process are invariant under conformal transformations that preserve the annulus $A_q$ (rotations and $q$-inversions).

3. Conditioning and Induced GAFs:
   Unlike the disk case where conditioning on zeros does not generate new GAF types, conditioning the GAF on $A_q$ at points $\alpha_1, \dots, \alpha_n$ induces a new GAF with covariance kernel $S_{A_q}(z, w; r \prod |\alpha_\ell|^2)$ (Proposition 1.2). This implies that the zero point process changes its statistical nature based on the configuration of conditioned zeros.

4. Correlation Behavior and Phases:
   The authors analyze the unfolded 2-correlation function $g_{A_q}(z, w; r)$.

   • Short Distance: The correlation is repulsive (negative), behaving as $(x-\sqrt{q})^2$ near the inner boundary, similar to DPPs (Proposition 1.5).
   • Long Distance: The behavior depends on the parameter $r$. There exists a critical value $r_0(q) \in (q, 1)$ such that:
     • If $r \in (q, r_0)$, the long-distance correlation is repulsive.
     • If $r \in (r_0, 1)$, attractive (positive) correlations emerge at large distances (Theorem 1.6).
   • Limit $q \to 0$: In the limit $q \to 0$, the process on the punctured disk $\mathbb{D}^\times$ becomes a PDPP $Z^r_{\mathbb{D}}$. This family interpolates between the Peres-Virág DPP ($r \to 0$) and the Peres-Virág DPP with a deterministic zero at the origin ($r \to \infty$). There is a critical value $r_c \approx 0.2846$ for this limit process. For $r < r_c$, the process is purely repulsive; for $r > r_c$, attractive interactions appear (Proposition 1.7).

Significance and Claims
The paper claims significance in generalizing the theory of GAF zeros from simply connected domains (disk) to multiply connected domains (annulus). The authors emphasize that this is not merely a replacement of determinants with per-determinants but involves fundamentally new structures:

1. Hierarchical Conditioning: The necessity of a series of new covariance kernels (weighted Szegő kernels) to describe conditioning, where the weight parameter encodes geometric information about the zeros.
2. Breakdown of Borchardt’s Identity: The identity $\text{perdet}(S) = \det(S^2)$, which ensures the DPP structure in the disk case, does not hold for the annulus. This leads to the PDPP structure and the coexistence of repulsive and attractive interactions.
3. Interpolation: The limit process $Z^r_{\mathbb{D}}$ provides a continuous interpolation between two known DPPs, offering a new perspective on the role of the origin in zero distributions.

The authors note that the study of PDPPs is nascent and list several open problems, including the geometric interpretation of the zero process on the annulus, the universality of the correlation decay index $\eta=4$, and potential extensions to Pfaffian-Hafnian point processes.