Wigner Random Matrices with Non-Symmetrically Distributed Entries

We show that the spectral radius of an N× N random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by 2 σ + o(N−6/11+ε), where σ² is the variance of the matrix entries and ε is an arbitrary small positive number. Our bound improves the earlier results by Z. Füredi and J. Komlós (1981), and Van Vu (2005).     

Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge

This is a continuation of our earlier paper (Tao and Vu, , 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (, 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697–733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.     

Spectral Properties of the k-Body Embedded Gaussian Ensembles of Random Matrices for Bosons

We consider m spinless Bosons distributed over l degenerate single-particle states and interacting through a k-body random interaction with Gaussian probability distribution (the Bosonic embedded k-body ensembles). We address the cases of orthogonal and unitary symmetry in the limit of infinite matrix dimension, attained either as l→∞ or as m→∞. We derive an eigenvalue expansion for the second moment of the many-body matrix elements of these ensembles. Using properties of this expansion, the supersymmetry technique, and the binary correlation method, we show that in the limit l→∞ the ensembles have nearly the same spectral properties as the corresponding Fermionic embedded ensembles. Novel features specific for Bosons arise in the dense limit defined as m→∞ with both k and l fixed. Here we show that the ensemble is not ergodic and that the spectral fluctuations are not of Wigner-Dyson type. We present numerical results for the dense limit using both ensemble unfolding and spectral unfolding. These differ strongly, demonstrating the lack of ergodicity of the ensemble. Spectral unfolding shows a strong tendency toward picket-fence-type spectra. Certain eigenfunctions of individual realizations of the ensemble display Fock-space localization.     

Analysis of Two Spectral Lines with Gaussian Profile by Using Photon Statistics. Comparison with Other Profile Shapes

In this paper we study the behavior of the errors involved in the determination of the parameters characterizing two spectral lines with Gaussian profile by means of the measurement of the intensity correlation function g⁽²⁾(τ) and by a multichannel system. The results obtained with both methods are compared. We also compare this results with those obtained when two lines with Lorentzian profile are analyzed.     

Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A. Lytova and L. Pastur (J. Stat. Phys. 134:147–159, 2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law. Moreover, if the marginal distributions satisfy the Poincaré inequality our results are valid for Lipschitz continuous test functions.     

Detecting Entanglement of States by Entries of Their Density Matrices

For any bipartite systems, a universal entanglement witness of rank-4 for pure states is obtained and a class of finite rank entanglement witnesses is constructed. In addition, a method of detecting entanglement of a state only by entries of its density matrix with respect to some product basis is obtained.     

Statistics of Gaussian Speckles with Enhanced Fluctuations

Some first-order statistics of speckle fields obeying the complex Gaussian probability density are investigated with the special emphasis on a possibility for the speckle contrast to exceed unity. It is shown that such enhanced fluctuations of speckle fields are generated when the probability density of the complex amplitude is non-circular with the vanishing mean. This means that Gaussian speckles having the contrast of unity are not necessarily fully developed speckles. Some practical situations producing such enhanced fluctuations in speckle fields are discussed.     

Estimates for Entries of Matrix Valued Functions of Infinite Matrices

Sharp upper estimates for the absolute values of entries of matrix valued functions of infinite matrices, as well as two sided estimates for the entries of matrix valued functions of infinite M-matrices (monotone matrices) are derived. They give us bounds for the lattice norms of matrix valued functions and positivity conditions for functions of M-matrices. In addition, some results on perturbations and comparison of matrix functions are proved. Applications of the obtained estimates to the Hille-Tamarkin matrices and differential equations are also discussed. This research was supported by the Kamea fund of the Israel     

System Driven by Correlated Gaussian Noises Related with Disorder

A system driven by correlated Gaussian noises related with disorder is investigated. The Fokker-Planck equation （FPE） for the system is derived. Using the FPE derived, some systems driven by correlated Gaussian noises related with disorder can be investigated for Brownian motors, nonequilibrium transition, resonant activation, stochastic resonance, and so on. We only give one example： i.e., using the FPE derived, we study the resonant activation for a single motor protein model with correlated noises related to disorder. Since the correlated noise related to disorder usually exists with the friction, for the temperature, and so on, our results have generic physical meanings for physics, chemistry, biology and other sciences.     

On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries

Journal of Statistical Physics - The paper is concerned with the asymptotic behavior of the correlation functions of the characteristic polynomials of non-Hermitian random matrices with independent...     

Spectral Analysis of Unitary Band Matrices

 This paper is devoted to the spectral properties of a class of unitary operators with a matrix representation displaying a band structure. Such band matrices appear as monodromy operators in the study of certain quantum dynamical systems. These doubly infinite matrices essentially depend on an infinite sequence of phases which govern their spectral properties. We prove the spectrum is purely singular for random phases and purely absolutely continuous in case they provide the doubly infinite matrix with a periodic structure in the diagonal direction. We also study some properties of the singular spectrum of such matrices considered as infinite in one direction only. Received: 29 April 2002 / Accepted: 7 August 2002 Published online: 20 January 2003 Communicated by B. Simon     

Spectral Problem of Block-rectangular Hierarchical Matrices

The spectral problem for matrices with a block-hierarchical structure is often considered in context of the theory of complex systems. In the present article, a new class of matrices with a block-rectangular non-symmetric hierarchical structure is introduced and the corresponding spectral problem is investigated. Using these results we study a model of error generation in information sequence where such block-rectangular hierarchical matrices appear in a natural way.     

Spectral Estimates for Periodic Jacobi Matrices

 We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ²(ℤ) of the form (Hψ) _n =a _n−1 ψ _n−1 +b _nψ _n +a _nψ _n+1 , where a _n=a _n+q and b _n=b _n+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of H. Received: 17 April 2002 / Accepted: 1 October 2002 Published online: 13 January 2003 Communicated by B. Simon     

Transmission of Correlated Gaussian Packets Through a Delta-Potential

We study the evolution of the most general initial Gaussian packet with nonzero correlation coefficient between the coordinate and momentum operators in the presence of a repulsive delta-potential barrier, using the known exact propagator of the time-dependent Schrödinger equation. For the initial packet localized far enough from the barrier, we define the transmission coefficient as the probability of discovering the particle in the whole semi-axis on the other side of the barrier. It appears that the asymptotical transmission coefficient (calculated in the large time limit) depends on two dimensionless parameters: the normalized ratio of the potential strength to the initial mean value of momentum and the ratio of the initial momentum dispersion to the initial mean value of momentum. For small values of the second parameter, the result is reduced to the well-known formula for the transparency of the delta barrier, obtained in the plane-wave approximation by solving the stationary Schrödinger equation. For large values of the second parameter, the transmission coefficient can be much larger than that calculated in the plane-wave approximation. For a fixed initial spread of the packet in the coordinate space, the initial correlation coefficient influences the transparency of the barrier only indirectly, through the increase in the initial momentum dispersion.     

Gaussian Multiplicative Chaos for Symmetric Isotropic Matrices

Motivated by isotropic fully developed turbulence, we define a theory of symmetric matrix valued isotropic Gaussian multiplicative chaos. Our construction extends the scalar theory developed by J.P. Kahane in 1985.     

Universality of the Edge Distribution of Eigenvalues of Wigner Random Matrices with Polynomially Decaying Distributions of Entries

We consider an ensemble of Wigner symmetric random matrices A_n={a_ij}, i,j=1, . . . ,n with matrix elements a_ij, being i.i.d. symmetrically distributed random variables We assume that and that for p>18. We prove that the distribution of the k (k=1,2, . . . ) largest (smallest) eigenvalues has a universal limit as n→∞ (the Tracy-Widom distribution).     

Spectral Properties of Amphiflavins in Polymer Matrices

The parameters characterizing the absorption, fluorescence and phosphorescence of 10-dodecylisoalloxazine (DIA), 10-octadecylisoalloxazine (OIA) 3-methyl-10-dodecylisoalloxazine (MDIA) embedded into polymeric matrices (polystyrene – PS, polymethacrylate methyl – PMM and poly(vinyl alcohol) – PVA) were studied. It was found that both the polarity of microenvironment and its structure influenced the changes of the spectral properties of the examined amphiflavins.