Power spectra and autocovariances of level spacings beyond the Dyson conjecture.

Introduced in the early days of random matrix theory, the autocovariances δI_{k}^{j}=cov(s_{j},s_{j+k}) of level spacings {s_{j}} accommodate detailed information on the correlations between individual eigenlevels. It was first conjectured by Dyson that the autocovariances of distant eigenlevels in the unfolded spectra of infinite-dimensional random matrices should exhibit a power-law decay δI_{k}^{j}≈-1/βπ^{2}k^{2}, where β is the symmetry index. In this Letter, we establish an exact link between the autocovariances of level spacings and their power spectrum, and show that, for β=2, the latter admits a representation in terms of a fifth Painlevé transcendent.

Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems.

We present a nonperturbative analysis of the power spectrum of energy level fluctuations in fully chaotic quantum structures. Focusing on systems with broken time-reversal symmetry, we employ a finite-N random matrix theory to derive an exact multidimensional integral representation of the power spectrum. The N→∞ limit of the exact solution furnishes the main result of this study-a universal, parameter-free prediction for the power spectrum expressed in terms of a fifth Painlevé transcendent.

Integrability of zero-dimensional replica field theories at β=1.

Building on insights from the theory of integrable lattices, the integrability is claimed for nonlinear replica σ models derived in the context of real symmetric random matrices. Specifically, the fermionic and the bosonic replica partition functions are proven to form a single (supersymmetric) Pfaff-KP hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density-density eigenvalue correlation function in the infinite-dimensional Gaussian orthogonal ensemble. Implications of the formalism outlined are briefly discussed.

Statistics of reflection eigenvalues in chaotic cavities with nonideal leads.

The scattering matrix approach is employed to determine a joint probability density function of reflection eigenvalues for chaotic cavities coupled to the outside world through both ballistic and tunnel point contacts. Derived under assumption of broken time-reversal symmetry, this result is further utilized to (i) calculate the density and correlation functions of reflection eigenvalues, and (ii) analyze fluctuations properties of the Landauer conductance for the illustrative example of asymmetric chaotic cavity. Further extensions of the theory are pinpointed.

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.

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.

Statistics of real eigenvalues in Ginibre's ensemble of random real matrices.

The integrable structure of Ginibre's 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.

Replica field theories, painlevé transcendents, and exact correlation functions.

Exact solvability is claimed for nonlinear replica sigma models derived in the context of random matrix theories. Contrary to other approaches reported in the literature, the framework outlined does not rely on traditional "replica symmetry breaking" but rests on a previously unnoticed exact relation between replica partition functions and Painlevé transcendents. While expected to be applicable to matrix models of arbitrary symmetries, the method is used to treat fermionic replicas for the Gaussian unitary ensemble (GUE), chiral GUE (symmetry classes A and AIII in Cartan classification) and Ginibre's ensemble of complex non-Hermitian random matrices.