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The forbidden region for random zeros: Appearance of quadrature domains
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2023-10-02 , DOI: 10.1002/cpa.22142
Alon Nishry 1 , Aron Wennman 1, 2
Affiliation  

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.

中文翻译:

随机零的禁区:正交域的出现

我们的主要发现是一方面正交域与高斯整体函数 (GEF) 的零点之间令人惊讶的相互作用。具体来说,考虑以罕见空洞事件为条件的 GEF ,即给定的大约旦域中不存在零。我们证明,在自然尺度极限下,包围孔的正交域作为禁区出现,其中零密度消失。此外,我们还描述了禁区为圆盘的孔类。随机零点和势论之间的连接由 Zeitouni-Zelditch 泛函的约束极值问题提供。为了解决这个问题,我们将其重新设计为一个看似新颖的障碍问题,其中解决方案被迫在孔内是谐波的。
更新日期:2023-10-02
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