Higher-order sum rules for zeros of the planar Gaussian entire function

Abstract

We establish marginal sum rules for the truncated correlation functions of zeros of the planar Gaussian entire function (GEF). For each fixed integer $k\ge2$, the integral in one coordinate converges absolutely and equals $-(k-1)$ times the truncated correlation of order $k-1$. This answers the marginal-identity question posed by Krishnapur and Yogeshwaran for the planar GEF. We also prove a criterion for simple point processes under local moment and absolute integrability assumptions. Suppose there is a sequence of compactly supported cutoffs between zero and one that are eventually equal to one on each compact set. If the variances of their linear statistics tend to zero, the marginal identities hold almost everywhere. Strong clustering and variance bounds for smooth linear statistics verify these assumptions for GEF zeros. The proof uses mixed cumulants and the partition formula relating ordinary and factorial cumulants.