TY - JOUR

T1 - Zeros of the i.i.d. Gaussian Laurent Series on an Annulus

T2 - Weighted Szegő Kernels and Permanental-Determinantal Point Processes

AU - Katori, Makoto

AU - Shirai, Tomoyuki

N1 - Funding Information:
The present study was stimulated by the useful discussion with Nizar Demni when the authors attended the workshop ‘Interactions between commutative and non-commutative probability’ held in Kyoto University, Aug 19–23, 2019. The present authors would like to thank Michael Schlosser for useful discussion on the elliptic extensions of determinantal and permanental formulas. The present work has been supported by the Grant-in-Aid for Scientific Research (C) (No.19K03674), (B) (No.18H01124), (S) (No.16H06338), and (A) (No.21H04432) of Japan Society for the Promotion of Science (JSPS).
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/6

Y1 - 2022/6

N2 - On an annulus Aq: = { 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 Aq 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 Szegő kernel of Aq 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↔ q/ z and the parameter change r↔ q2/ r. When r= q they are invariant under conformal transformations which preserve Aq. 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→ 0 , a simpler but still non-trivial PDPP is obtained on the unit disk D. We observe that the limit PDPP indexed by r∈ (0 , ∞) can be regarded as an interpolation between the determinantal point process (DPP) on D studied by Peres and Virág (r→ 0) and that DPP of Peres and Virág with a deterministic zero added at the origin (r→ ∞).

AB - On an annulus Aq: = { 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 Aq 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 Szegő kernel of Aq 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↔ q/ z and the parameter change r↔ q2/ r. When r= q they are invariant under conformal transformations which preserve Aq. 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→ 0 , a simpler but still non-trivial PDPP is obtained on the unit disk D. We observe that the limit PDPP indexed by r∈ (0 , ∞) can be regarded as an interpolation between the determinantal point process (DPP) on D studied by Peres and Virág (r→ 0) and that DPP of Peres and Virág with a deterministic zero added at the origin (r→ ∞).

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U2 - 10.1007/s00220-022-04365-2

DO - 10.1007/s00220-022-04365-2

M3 - Article

AN - SCOPUS:85127384128

VL - 392

SP - 1099

EP - 1151

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -