Eugene Kanzieper

Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

Abstract In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability pn,k to find exactly k real eigenvalues in the spectrum of an n×n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined.

Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices

The integrable structure of Ginibres orthogonal ensemble of random matrices is looked at through the prism of the probability p(n,k) to find exactly k real eigenvalues in the spectrum of an n x n real asymmetric Gaussian random matrix. The exact solution for the probability function p(n,k) is presented, and its remarkable connection to the theory of symmetric functions is revealed. An extension of the Dyson integration theorem is a key ingredient of the theory presented.

Integrable theory of quantum transport in chaotic cavities.

The problem of quantum transport in chaotic cavities with broken time-reversal symmetry is shown to be completely integrable in the universal limit. This observation is utilized to determine the cumulants and the distribution function of conductance for a cavity with ideal leads supporting an arbitrary number n of propagating modes. Expressed in terms of solutions to the fifth Painlevé transcendent and/or the Toda lattice equation, the conductance distribution is further analyzed in the large-n limit that reveals long exponential tails in the otherwise Gaussian curve.

Statistics of thermal to shot noise crossover in chaotic cavities

Recently formulated integrable theory of quantum transport (Osipov and Kanzieper, 2008 Phys. Rev. Lett. 101 176804) is extended to describe sample-to-sample fluctuations of the noise power in chaotic cavities with broken time-reversal symmetry. Concentrating on the universal transport regime, we determine dependence of the noise power cumulants on the temperature, applied bias voltage and the number of propagating modes in the leads. Intrinsic connection between statistics of thermal to shot noise crossover and statistics of Landauer conductance is revealed and briefly discussed.

Non-Hermitean Wishart random matrices "I…

A non-Hermitean extension of paradigmatic Wishart random matrices is introduced to set up a theoretical framework for statistical analysis of (real, complex, and real-quaternion) stochastic time series representing two “remote” complex systems. The first paper in a series provides a detailed spectral theory of non-Hermitean Wishart random matrices composed of complex valued entries. The great emphasis is placed on an asymptotic analysis of the mean eigenvalue density for which we derive, among other results, a complex-plane analog of the Marcenko–Pastur law. A surprising connection with a class of matrix models previously invented in the context of quantum chromodynamics is pointed out.

Correlations of RMT characteristic polynomials and integrability: Hermitean matrices

Abstract Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of τ functions, we (i) identify a zoo of hierarchical relations satisfied by τ functions in an abstract infinite-dimensional space and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasis is placed on the phenomenon of fermionic–bosonic factorisation of random-matrix-theory correlation functions.

Are Bosonic Replicas Faulty

Motivated by the ongoing discussion about a seeming asymmetry in the performance of fermionic and bosonic replicas, we present an exact, nonperturbative approach to both fermionic and bosonic zero-dimensional replica field theories belonging to the broadly interpreted beta=2 Dyson symmetry class. We then utilize the formalism developed to demonstrate that the bosonic replicas do correctly reproduce the microscopic spectral density in the QCD-inspired chiral Gaussian unitary ensemble. This disproves the myth that the bosonic replica field theories are intrinsically faulty.

What is the probability that a large random matrix has no real eigenvalues

We study the large-

A note on the Pfaffian integration theorem

n

Random matrix theory of quantum transport in chaotic cavities with nonideal leads

limit of the probability