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

Abstract

On an annulus A q := {z ∈ C : q < |z| < 1} with a fixed q ∈ (0, 1), we study a Gaussian analytic function (GAF) and its zero set which defines a point process on 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 given by the weighted Szegő kernel of A q with the weight parameter r studied by Mccullough and Shen. The GAF and the zero-point process have symmetry associated with the q-inversion of coordinate z ↔ q/z and the parameter change r ↔ q 2 /r, and when r = q they are invariant under conformal transformations which preserve A q . Conditioning the GAF by giving 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 given zeros. Then we 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 → 0, a simpler but still non-trivial PDPP is obtained on a punctured unit disk D × := D \ {0}. In the further limit r → 0 the present PDPP is reduced to the determinantal point process on D studied by Peres and Virág.

Polyanalytic reproducing Kernels on the quantized annulus

“…By direct calculations, one checks that the poly-analyticity property with respect to ∂ z fails for the basis elements of E m Ω 1,R , m = 0 while it still holds true for the invariant Cauchy-Riemann operator. Finally, we would like to point out the recent preprint [16] where the authors relate zeroes of Laurent series with Gaussian coefficients to a hyper-determinantal point process governed by the Szegö kernel of the annulus (see also [15] for other connections of DPP to elliptic functions).…”

Universality for outliers in weakly confined Coulomb-type systems

“…In addition, the proof of Theorem 1.6 can be adapted to the setting of a finitely connected uncharged region. This is related to [20] which deals with zeros of random Laurent series with i.i.d. coefficients.…”

Expected number of zeros of random power series with finitely dependent Gaussian coefficients

“…The studies around this Gaussian analytic function (GAF) has been developing in several directions (cf. [1,3,5,8,10,11]), however, it seems that there are relatively few works on zeros of random power series with dependent Gaussian coefficients. Recently, Mukeru, Mulaudzi, Nazabanita and Mpanda studied the zeros of Gaussian random power series f H (z) on the unit disk with coefficients Ξ (H) = {ξ (H) k } ∞ k=0 being a fractional Gaussian noise (fGn) with Hurst index 0 ≤ H < 1.…”