Fekete Points as Norming Sets for Random Holomorphic Sections

Abstract

We prove that Fekete configurations associated with a positive Hermitian line bundle over a compact Kähler manifold are asymptotically exact norming sets for random holomorphic sections. For every fixed $a>0$, the ratio of the supremum norm of a degree-$k$ section to its maximum on the degree-$\lceil ak\rceil$ Fekete configuration converges to one in probability. This holds for any background Kähler form: Fekete equidistribution provides enough sampling points near the maximum of the limiting pointwise variance. We obtain polynomial upper bounds for the tail of the ratio and almost-sure norm comparisons. In contrast, when the limiting variance is nonconstant, we construct separated arrays of cardinality comparable to the section-space dimension for which the ratio converges to a constant strictly greater than one. General separated arrays still satisfy probabilistic norming inequalities with degree-independent constants, and every such array is asymptotically exact if and only if the volume form is proportional to $c_1(L,h{)}^n$. We also prove a discrete $L^2$ norm comparison with exponentially small exceptional probability. The norm comparisons and convergence statements hold in both the Gaussian and spherical ensembles.

arXiv commentv2:Substantially revised and retitled. The previous claim that (Λ_k/N_k\to1) has been withdrawn. New results include asymptotically exact probabilistic norming, quantitative tail estimates, and extensions to general separated arrays. 24 pages

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