Random Morse functions and spectral geometry
Random Morse functions and spectral geometry
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随机莫尔斯函数和谱几何
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发表时间：
2012
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通讯作者：
L. Nicolaescu
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作者：
L. Nicolaescu
研究了黎曼流形$（M^m，g）$上的随机莫尔斯函数，定义为度量$g$的拉普拉斯算子的特征函数的随机高斯加权叠加.随机性由固定的Schwartz函数$w$和小参数$vareps>0$确定。我们首先证明，作为$vareps， o 0$这个随机函数的临界值的预期分布接近$mathbb{R}$上的普适测度，与$g$无关，可以用随机$（m+1）的高斯维格纳系综的本征值的预期分布来明确描述。 imes（m+1）$对称矩阵与此相反，我们证明了度量$g$和它的曲率是由统计的随机函数的海森小$varepp $。
We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a small parameter $varepsilon>0$. We first prove that as $varepsilon o 0$ the expected distribution of critical values of this random function approaches a universal measure on $mathbb{R}$, independent of $g$, that can be explicitly described in terms the expected distribution of eigenvalues of the Gaussian Wigner ensemble of random $(m+1) imes (m+1)$ symmetric matrices. In contrast, we prove that the metric $g$ and its curvature are determined by the statistics of the Hessians of the random function for small $varepsilon$.