On the ubiquity of the Cauchy distribution in spectral problems
On the ubiquity of the Cauchy distribution in spectral problems
复制标题
论谱问题中柯西分布的普遍性
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
S. Warzel
中科院分区：
文献类型：
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作者：
M. Aizenman;S. Warzel
研究了上半平面的解析映射Herglotz-Pick （HP）类随机函数实数点上值的分布。在温和平稳性假设下，具有奇异谱的HP函数的个体值具有柯西型分布。该表述适用于随机算子的对角矩阵元素，无论是否存在水平斥力，即适用于随机矩阵和泊松型谱。
We consider the distribution of the values at real points of random functions which belong to the Herglotz–Pick (HP) class of analytic mappings of the upper half plane into itself. It is shown that under mild stationarity assumptions the individual values of HP functions with singular spectra have a Cauchy type distribution. The statement applies to the diagonal matrix elements of random operators, and holds regardless of the presence or not of level repulsion, i.e. applies to both random matrix and Poisson-type spectra.