The forbidden region for random zeros: Appearance of quadrature domains

Alon Nishry*, Aron Wennman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1766-1849
Number of pages84
JournalCommunications on Pure and Applied Mathematics
Issue number3
StatePublished - Mar 2024


FundersFunder number
European Research Council692616
Israel Science Foundation1903/18
Knut och Alice Wallenbergs Stiftelse2017.0389


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