On the Distribution of the Critical Values of Random Spherical Harmonics
On the Distribution of the Critical Values of Random Spherical Harmonics
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DOI：
10.1007/s12220-015-9668-5
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发表时间：
2014-09
期刊：
影响因子：
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通讯作者：
Valentina Cammarota;D. Marinucci;I. Wigman
中科院分区：
文献类型：
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作者：
Valentina Cammarota;D. Marinucci;I. Wigman
我们研究高能极限下随机球谐函数的临界点和极值的极限分布。特别是，我们首先推导极值和鞍点的密度函数；然后，我们提供了方差的分析表达式，并且我们表明，高能量极限下的经验测量与预期值的收敛程度较弱。我们的论点需要仔细研究 Kac-Rice 公式在非标准情况下的有效性，这需要分析过程的一阶和二阶导数的协方差矩阵的简并性。
We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high-energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for the variances and we show that the empirical measures in the high-energy limits converge weakly to their expected values. Our arguments require a careful investigation of the validity of the Kac–Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed.