Asymptotic distribution of random zeros outside the support of the symbol

Determine the asymptotic distribution, as the tensor power p→∞, of the zero current [Div(S_{f,p})] of the random L-holomorphic sections S_{f,p}=T_{f,p}S_p on a compact Hermitian complex manifold (with positive line bundle) on the region outside the essential support of the symbol f; in particular, characterize the limiting behavior of [Div(S_{f,p})] on X\ess.supp f.

Background

In the compact case, the random section is given by S_{f,p}=T_{f,p}S_p, where S_p is a Gaussian holomorphic section determined by the Bergman space H^0(X,L^p\otimes E). The paper proves equidistribution on the support of f and presents simulations on \mathbb{CP}^1 illustrating strong agreement inside the support, while highlighting markedly different behavior outside.

Despite extensive near-diagonal kernel asymptotics and concentration results on the support, the limiting distribution of random zeros outside the support of f is not derived. The authors explicitly point out that this problem remains open even in the compact setting, motivating further analysis of the interaction between Toeplitz operators and the geometry of the complement of the support.