Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices

TL;DR

This paper develops an integrable theory for correlation functions of characteristic polynomials in invariant non-Gaussian Hermitian random matrix ensembles, linking tau-functions to nonlinear differential equations and exploring fermionic-bosonic factorization.

Contribution

It introduces a hierarchical framework connecting tau-functions to correlation functions, enabling new differential equation descriptions for non-Gaussian Hermitian matrices.

Findings

Hierarchical relations for tau-functions identified

Differential equations for correlation functions derived

Fermionic-bosonic factorization phenomena analyzed

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.