Characteristic Polynomials of Complex Random Matrix Models

Abstract

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit.

LaTeX, 22 pages; typos corrected, comment added

References in corpus (9)

• PP-wave string interactions from perturbative Yang-Mills theory
• Replica Limit of the Toda Lattice Equation
• Replica field theories, Painleve transcendents, and exact correlation functions
• Large scale correlations in normal and general non-Hermitian matrix ensembles
• Collective String Field Theory of Matrix Models in the BMN Limit
• New applications of non-hermitian random matrices
• BMN Correlators by Loop Equations
• Microscopic universality of complex matrix model correlation functions at weak non-Hermiticity
• Matrix model correlation functions and lattice data for the QCD Dirac operator with chemical potential

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