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

Abstract

On an annulus $${{\mathbb {A}}}_q :=\{z \in {{\mathbb {C}}}: q< |z| < 1\}$$ A q : = { z ∈ C : q < | z | < 1 } with a fixed $$q \in (0, 1)$$ q ∈ ( 0 , 1 ) , we study a Gaussian analytic function (GAF) and its zero set which defines a point process on $${{\mathbb {A}}}_q$$ 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 $$r >0$$ r > 0 is identified with the weighted Szegő kernel of $${{\mathbb {A}}}_q$$ 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 $$z \leftrightarrow q/z$$ z ↔ q / z and the parameter change $$r \leftrightarrow q^2/r$$ r ↔ q 2 / r . When $$r=q$$ r = q they are invariant under conformal transformations which preserve $${{\mathbb {A}}}_q$$ 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 $$q \rightarrow 0$$ q → 0 , a simpler but still non-trivial PDPP is obtained on the unit disk $${\mathbb {D}}$$ D . We observe that the limit PDPP indexed by $$r \in (0, \infty )$$ r ∈ ( 0 , ∞ ) can be regarded as an interpolation between the determinantal point process (DPP) on $${{\mathbb {D}}}$$ D studied by Peres and Virág ($$r \rightarrow 0$$ r → 0 ) and that DPP of Peres and Virág with a deterministic zero added at the origin ($$r \rightarrow \infty $$ r → ∞ ).

Point Processes and Multiple SLE/GFF Coupling

“…Here we have reported our work to generalize the above to a family of GAFs and their zero point processes on the annulus A q . We can prove the following [36].…”

“…The kernel (2.37) with a parameter r > 0 is called the weighted Szegő kernel of A q and H 2 r (A q ) is the reproducing kernel Hilbert space (RKHS) with respect to S Aq (•, •; r) [52]. We call r the weight parameter [36]. Notice that (2.37) implies that S Aq (z, z; r) is a monotonically decreasing function of the weight parameter r ∈ (0, ∞) for each fixed z ∈ A q .…”

“…that is, perdet M denotes per M multiplied by det M . Note that perdet is a special case of hyperdeterminants introduced by Gegenbauer following Cayley [36]. If M is a positive semidefinite Hermitian matrix, then per M ≥ det M ≥ 0, and hence perdet M ≥ 0 by the definition (2.67).…”

“…In addition the proportionality between the square of the Szegő kernel and the Bergman kernel (2.79), which holds as (2.79) on D, is no longer valid for the point processes on A q The following are verified [36]. A CONS for the Bergman space on A q is given by { e (q) n (z)} n∈Z where we set…”

“…We can prove the equality [36] S Aq (z, w) 2 = K Aq (z, w) + a(q) zw , z, w ∈ A q , (2.86) where a(q) = −2 n∈N (−1) n nq n 1 − q 2n + 1 2 log q .…”