Boris A. Khoruzhenko

Asymptotic properties of large random matrices with independent entries

We study the normalized trace gn(z)=n−1 tr(H−zI)−1 of the resolvent of n×n real symmetric matrices H=[(1+δjk)Wjk√n]j,k=1n assuming that their entries are independent but not necessarily identically distributed random variables. We develop a rigorous method of asymptotic analysis of moments of gn(z) for | Iz|≥η0 where η0 is determined by the second moment of Wjk. By using this method we find the asymptotic form of the expectation E{gn(z)} and of the connected correlator E{gn(z1)gn(z2)}−E{gn(z1)}E{gn (z2)}. We also prove that the centralized trace ngn(z)−E{ngn(z)} has the Gaussian distribution in the limit n=∞. Based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles.

Almost Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre Eigenvalue Statistics

By using the method of orthogonal polynomials, we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner-Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures [as, e.g., spectral form factor, number variance, and small distance behavior of the nearest neighbor distance distribution p(s) ] are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior p(s){proportional_to}s{sup 5/2} for some parameter values. {copyright} {ital 1997} {ital The American Physical Society}

Almost-Hermitian random matrices: eigenvalue density in the complex plane

Abstract We consider an ensemble of large non-Hermitian random matrices of the form H + i A s , where H and A s are Hermitian statistically independent random N × N matrices. We demonstrate the existence of a new nontrivial regime of weak non-Hermiticity characterized by the condition that the average of N Tr A s 2 is of the same order as that of Tr H 2 when N → ∞. We find explicitly the density of complex eigenvalues for this regime in the limit of infinite matrix dimension. The density determines the eigenvalue distribuyion in the crossover regime between random Hermitian matrices whose real eigenvalues are distributed according to the Wigner semi-circle law and random complex matrices whose eigenvalues are distributed in the complex plane according to the so-called “elliptic law”.

Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble

The goal of this paper is to establish a relation between characteristic polynomials of N×N GUE random matrices H as N→∞, and Gaussian processes with logarithmic correlations. We introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of D_N(z)=−log|det(H−zI)| on mesoscopic scales as N→∞. By employing a Fourier integral representation, we use this to prove a continuous analogue of a result by Diaconis and Shahshahani [J. Appl. Probab. 31A (1994) 49–62]. On the macroscopic scale, D_N(x) gives rise to yet another type of Gaussian process with logarithmic correlations. We give an explicit construction of the latter in terms of a Chebyshev–Fourier random series.

Systematic Analytical Approach to Correlation Functions of Resonances in Quantum Chaotic Scattering

We solve the problem of resonance statistics in systems with broken time-reversal invariance by deriving the joint probability density of all resonances in the framework of a random matrix approach and calculating explicitly all n-point correlation functions in the complex plane. As a by-product, we establish the Ginibre-like statistics of resonances for many open channels. Our method is a combination of Itzykson-Zuber integration and a variant of nonlinear

Induced Ginibre ensemble of random matrices and quantum operations

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Systematic approach to statistics of conductance and shot-noise in chaotic cavities

model and can be applied when the use of orthogonal polynomials is problematic.

Truncations of random orthogonal matrices

A generalization of the Ginibre ensemble of non-Hermitian random square matrices is introduced. The corresponding probability measure is induced by the ensemble of rectangular Gaussian matrices via a quadratization procedure. We derive the joint probability density of eigenvalues for such an induced Ginibre ensemble and study various spectral correlation functions for complex and real matrices, and analyse universal behaviour in the limit of large dimensions. In this limit, the eigenvalues of the induced Ginibre ensemble cover uniformly a ring in the complex plane. The real induced Ginibre ensemble is shown to be useful to describe the statistical properties of evolution operators associated with random quantum operations for which the dimensions of the input state and the output state do differ.

The Thouless formula for random non-Hermitian Jacobi matrices

Applying random matrix theory to quantum transport in chaotic cavities, we develop a powerful method for computing the moments of the conductance and shot-noise including their joint moments of arbitrary order and at any number of open channels. Our approach is based on the Selberg integral theory combined with the theory of symmetric functions and is applicable equally well for systems with and without time-reversal symmetry. We also compute higher-order cumulants and perform their detailed analysis. In particular, we establish an explicit form of the leading asymptotic of the cumulants in the limit of the large channel numbers. We derive further a general Pfaffian representation for the corresponding distribution functions. The Edgeworth expansion based on the first four cumulants is found to reproduce fairly accurately the distribution functions in the bulk even for a small number of channels. As the latter increases, the distributions become Gaussian-like in the bulk but are always characterized by a power-law dependence near their edges of support. Such asymptotics are determined exactly up to linear order in distances from the edges, including the corresponding constants.

The localization of surface states: an exactly solvable model

Statistical properties of nonsymmetric real random matrices of size M, obtained as truncations of random orthogonal N×N matrices, are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong nonorthogonality, M/N=const, the behavior typical to real Ginibre ensemble is found. In the case M=N-L with fixed L, a universal distribution of resonance widths is recovered.