RMT特征多项式与Hermite矩阵可积性的关系
Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
摘要：
给出了厄米特随机矩阵不变非高斯系综特征多项式的关联函数的可积理论。通过将感兴趣的关联函数嵌入到更一般的τ函数理论中，我们（i）确定了一个抽象的无限维空间中τ函数所满足的层次关系的动物园，以及（ii）提出了一种技术，将这些关系转化为描述物理谱空间中特征多项式的关联函数的层次结构的非线性微分方程。这种形式主义的影响，费米子，玻色子和超对称变化的零维副本场论进行了详细讨论。一个特别强调的是放在随机矩阵理论关联函数的费米子玻色子因式分解的现象。
英文摘要：
Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of tau-functions, we (i) identify a zoo of hierarchical relations satisfied by tau-functions in an abstract infinite-dimensional space, and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasis is placed on the phenomenon of fermionic-bosonic factorisation of random-matrix-theory correlation functions.
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