论文标题
节点域数量的波动
Fluctuations in the number of nodal domains
论文作者
论文摘要
我们表明，n n的正面球形谐波的二维高斯集合的零集合的连接组件数量的差异是n的正增长。该证明不使用球形谐波的特殊特性，并适用于在二维球体上具有足够规则的高斯随机函数的任何足够规则的合奏，并且相对于球体的异构体，分布不变。 我们的论点将淋巴结线数量的波动与第四级平面图上的随机循环集合中的波动联系起来，这可以看作是迈向Bogomolny-Schmit启发式方法的依据的一步。
We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step towards justification of the Bogomolny-Schmit heuristics.
