利用自由概率的超对称延拓放大局部水平统计量
Zooming in on local level statistics by supersymmetric extension of free probability
摘要：
我们考虑具有约束势NV的厄米特NxN矩阵H的酉系综，其中V是解析一致凸的。从Zinn-Justin，Collins，Guionnet和Maida的工作中可以知道，有限秩傅立叶变量K的特征函数的大N极限由Voiculescu R变换确定，这是自由概率论中的一个关键对象。超越这些结果，我们认为，同样成立时，有限秩算子K的形式，是所需的韦格纳-埃菲托夫超对称方法的积分交换和反交换变量。这种见解导致了一种有效的新技术，用于研究当地的统计数据，例如，水平相关性我们说明了新的技术，证明在随机矩阵模型的随机散射的普遍性。
英文摘要：
We consider unitary ensembles of Hermitian NxN matrices H with a confining potential NV where V is analytic and uniformly convex. From work by Zinn-Justin, Collins, and Guionnet and Maida it is known that the large-N limit of the characteristic function for a finite-rank Fourier variable K is determined by the Voiculescu R-transform, a key object in free probability theory. Going beyond these results, we argue that the same holds true when the finite-rank operator K has the form that is required by the Wegner-Efetov supersymmetry method of integration over commuting and anti-commuting variables. This insight leads to a potent new technique for the study of local statistics, e.g., level correlations. We illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering.
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