On the Structure of Correlation Functions in the Normal Matrix Model

Abstract:

We study the structure of the normal matrix model (NMM). We show that all correlation functions of the model with an axially symmetric potential can be expressed in terms of a single holomorphic function of one variable. This observation is used to demonstrate the exact solvability of the model. The two-point correlation function is calculated in the continuum limit. The answer is proven to be universal, i.e. potential independent up to a change of the scale. In connection with NMM with a general polynomial potential we have developed a two-dimensional free fermion formalism and constructed a family of completely integrable hierarchies of non-linear differential equations, which we call the extended-KP(N) hierarchies. The well-known KP hierarchy is a lower-dimensional reduction of this family. The extended-KP(1) hierarchy contains the (2+1)-dimensional Burgers equations. The partition function of the (N×N) NMM is the τ function of the extended-KP(N) hierarchy which is invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane.