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.
Keywords
Cite
@article{arxiv.cond-mat/0312006,
title = {Exact replica treatment of non-Hermitean complex random matrices},
author = {Eugene Kanzieper},
journal= {arXiv preprint arXiv:cond-mat/0312006},
year = {2007}
}
Comments
Dedicated to the memory of Professor Iya Ipatova; (v3: published version, references updated)