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29 March 2024
 
  » arxiv » cond-mat/0312006

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Exact replica treatment of non-Hermitean complex random matrices
Eugene Kanzieper ;
Date 29 Nov 2003
Journal Published in: Frontiers in Field Theory, edited by O. Kovras, Ch. 3, pp. 23 -- 51 (Nova Science Publishers, NY 2005). ISBN: 1-59454-127-2
Subject Disordered Systems and Neural Networks; Statistical Mechanics; Mathematical Physics; Exactly Solvable and Integrable Systems | cond-mat.dis-nn cond-mat.stat-mech hep-th math-ph math.MP nlin.SI
AbstractRecently 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.
Source arXiv, cond-mat/0312006
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