Correlations of RMT characteristic polynomials and integrability: Hermitean matrices

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 τ functions, we (i) identify a zoo of hierarchical relations satisfied by τ 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.