TY - JOUR

T1 - The forbidden region for random zeros

T2 - Appearance of quadrature domains

AU - Nishry, Alon

AU - Wennman, Aron

N1 - Publisher Copyright:
© 2023 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.

PY - 2024/3

Y1 - 2024/3

N2 - Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zeros of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero density vanishes. Moreover, we give a description of the class of holes for which the forbidden region is a disk. The connecting link between random zeros and potential theory is supplied by a constrained extremal problem for the Zeitouni-Zelditch functional. To solve this problem, we recast it in terms of a seemingly novel obstacle problem, where the solution is forced to be harmonic inside the hole.

AB - Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zeros of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero density vanishes. Moreover, we give a description of the class of holes for which the forbidden region is a disk. The connecting link between random zeros and potential theory is supplied by a constrained extremal problem for the Zeitouni-Zelditch functional. To solve this problem, we recast it in terms of a seemingly novel obstacle problem, where the solution is forced to be harmonic inside the hole.

UR - http://www.scopus.com/inward/record.url?scp=85173428590&partnerID=8YFLogxK

U2 - 10.1002/cpa.22142

DO - 10.1002/cpa.22142

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AN - SCOPUS:85173428590

SN - 0010-3640

VL - 77

SP - 1766

EP - 1849

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 3

ER -