This note concerns the asymptotics of the expected total Betti numbers of the nodal set for an important class of Gaussian ensembles of random fields on Riemannian manifolds. By working with the limit random field defined on the Euclidean space we were able to obtain a locally precise asymptotic result, though due to the possible positive contribution of large percolating components this does not allow us to infer a global result. As a by-product of our analysis, we refine the lower bound of Gayet and Welschinger for the important Kostlan ensemble of random polynomials and its generalisation to Kähler manifolds.

中文翻译：

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关于随机域节点集的期望 Betti 数

本注释涉及黎曼流形上一类重要的随机场高斯系综的节点集的预期总 Betti 数的渐近性。通过使用定义在欧几里德空间上的极限随机场，我们能够获得局部精确的渐近结果，尽管由于大渗透分量可能的积极贡献，这不允许我们推断全局结果。作为我们分析的副产品，我们改进了 Gayet 和 Welschinger 的下界，用于重要的随机多项式 Kostlan 系综及其对 Kähler 流形的推广。