Exact replica treatment of non-Hermitean complex random matrices

Abstract

Recently discovered exact integrability of zero-dimensional replica field theories [E. Kanzieper, Phys. Rev. Lett. 89, 250201 (2002)] is examined in the context of Ginibre Unitary Ensemble of non-Hermitean random matrices (GinUE). In particular, various nonperturbative fermionic replica partition functions for this random matrix model are shown to belong to a positive, semi-infinite Toda Lattice Hierarchy which, upon its Painleve reduction, yields exact expressions for the mean level density and the density-density correlation function in both bulk of the complex spectrum and near its edges. Comparison is made with an approximate treatment of non-Hermitean disordered Hamiltonians based on the "replica symmetry breaking" ansatz. A difference between our replica approach and a framework exploiting the replica limit of an infinite (supersymmetric) Toda Lattice equation is also discussed.

arXiv commentDedicated to the memory of Professor Iya Ipatova; (v3: published version, references updated)

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Cited by in corpus (4)

• Universal microscopic correlation functions for products of truncated unitary matrices
• A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble
• The Interpolating Airy Kernels for the beta=1 and beta=4 Elliptic Ginibre Ensembles
• Non-differentiability of the effective potential and the replica symmetry breaking in the random energy model