作为随机全纯截面范数集的Fekete点
Fekete Points as Norming Sets for Random Holomorphic Sections
AI总结：
该研究证明紧Kähler流形上正Hermitian线丛的Fekete构型是随机全纯截面的渐近精确范数集，还给出相关范数比较与收敛结论，构造了反例并推导了相关条件。
摘要：
我们证明，紧Kähler流形上正Hermitian线丛对应的Fekete构型，是随机全纯截面的渐近精确范数集。对任意固定的a>0，次数为k的截面的上确界范数，与其在次数为⌈ak⌉的Fekete构型上的最大值之比，依概率收敛于1。该结论对任意背景Kähler形式成立：Fekete等分布在极限逐点方差的最大值附近提供了足够的采样点。我们得到该比值尾部的多项式上界，以及几乎必然的范数比较结果。相反，当极限方差非恒定时，我们构造了基数与截面空间维数相当的分离阵列，其比值收敛于严格大于1的常数。一般分离阵列仍满足与次数无关的概率范数不等式，且此类阵列渐近精确当且仅当体积形式与c₁(L,h)ⁿ成比例。我们还证明了离散L²范数比较，其例外概率指数级小。范数比较与收敛结论在高斯和球形系综中均成立。
英文摘要：
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.
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