Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices

Abstract

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.

62 pages, 1 table, published version (typos corrected)

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Cited by in corpus (9)

• The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation
• Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
• On characteristic polynomials for a generalized chiral random matrix ensemble with a source
• On Airy Solutions of the Second Painlevé Equation
• Power spectrum of the circular unitary ensemble
• On Integrable Ermakov-Painlevé IV Systems
• Integrable Aspects of Universal Quantum Transport in Chaotic Cavities
• On a distinguished family of random variables and Painlevé equations
• Integrability of zero-dimensional replica field theories at beta=1