Leonid Pastur

Spectra of random and almost-periodic operators

I. Metrically Transitive Operators.- 1 Basic Definitions and Examples.- 1.A Random Variables, Functions and Fields.- 1.B Random Vectors and Operators.- l.C Metrically Transitive Random Fields.- l.D Metrically Transitive Operators.- 2 Simple Spectral Properties of Metrically Transitive Operators.- 2.A Deficiency Indices.- 2.B Nonrandomnessofthe Spectrum and of its Components.- 2.C Nonrandomness of Multiplicities.- Problems.- II. Asymptotic Properties of Metrically Transitive Matrix and Differential Operators.- 3 Review of Basic Results.- 4 Matrix Operators on ?2 (Zd).- 4.A Essential Self-Adjointness.- 4.B Existence of the Integrated Density of States and Other Ergodic Properties.- 4.C Simple Properties of the Integrated Density of States and of the Spectra of Metrically Transitive Matrix Operators.- 4.D Location of the Spectrum.- 5 Schrodinger Operators and Elliptic Differential Operators on L2(Rd).- 5.A Criteria for Essential Self-Adjointness.- 5.B Ergodic Properties.- 5.C Some Properties of the Integrated Density of States.- 5.D Location of the Spectrum of a Metrically Transitive Schrodinger Operator.- Problems.- III. Integrated Density of States in One-Dimensional Problems of Second Order.- 6 The Oscillation Theorem and the Integrated Density of States.- 6. A The Phase and the Existence of the Integrated Density of States.- 6.B Simplest Asymptotics of the Integrated Density of States at the Edges of the Spectrum.- 6.C Schrodinger Operator with Markov Potential.- 6.D The Brownian Motion Model.- 6.E Jacobi Matrices with Independent and Markov Coefficients.- 6.F Smoothness of N (?) Special Energies.- 7 Examples of Calculation of the Integrated Density of States.- 7.A The Kronig-Penny Stochastic Model.- 7.B Random Jacobi Matrices.- Problems.- IV. Asymptotic Behavior of the Integrated Density of States at Spectral Boundaries in Multidimensional Problems.- 8 Stable Boundaries.- 9 Fluctuation Boundaries: General Discussion and Classical Asymptotics.- 9.A Introduction and Heuristic Discussion.- 9.B Simplest Bounds. Gaussian and Negative Poisson Potentials.- 9.C Generalized Poisson Potential.- 10 Fluctuation Boundaries: Quantum Asymptotics.- 10.A The Lifshitz Exponent.- 10.B Generalized Poisson Potential with a Nonnegative, Rapidly Decreasing Function.- 10.C Smoothed Square of a Gaussian Random Field.- Problems.- V. Lyapunov Exponents and the Spectrum in One Dimension.- 11 Existence and Properties of Lyapunov Exponents.- 11.A The Multiplicative Ergodic Theorem and the Existence of Lyapunov Exponents.- 11.B The Lyapunov Exponent and the Integrated Density of States.- 11.C Simplest Asymptotic Formulas and Estimates for Lyapunov Exponents.- 12 Lyapunov Exponents and the Absolutely Continuous Spectrum.- 12.A Basic Facts About the Spectrum of One-Dimensional Operators of the Second Order.- 12.B Lyapunov Exponents and the Absolutely Continuous Spectrum.- 12.C Multiplicity of the Spectrum.- 12.D Deterministic Potentials.- 12.E Some Inverse Problems.- 13 Lyapunov Exponents and the Point Spectrum.- 13.A Heuristic Discussion.- 13.B Conditions for Positive Lyapunov Exponents to Imply a Pure Point Spectrum.- Problems.- VI. Random Operators.- 14 The Lyapunov Exponent of Random Operators in One Dimension.- 14.A Positiveness of the Lyapunov Exponent.- 14.B Asymptotic Formulas for the Lyapunov Exponent.- 15 The Point Spectrum of Random Operators.- 15.A The Pure Point Spectrum in One Dimension.- 15.B Other One-Dimensional Results.- 15.C The Point Spectrum in Multidimensional Problems.- Problems.- VII. Almost-Periodic Operators.- 16 Smooth Quasi-Periodic Potentials.- 16.A The Integrated Density of States and the Gap Labeling Theorem.- 16.B Absolutely Continuous Spectrum.- 16.C Lower Bounds of Solutions and Absence of a Point Spectrum.- 16.D Lower Bounds for the Lyapunov Exponent and Absence of an Absolutely Continuous Spectrum in the Discrete Case.- 16.E Point Spectrum of Almost-Periodic Operators.- 16.F The Almost-Mathieu Operator.- 17 Limit-Periodic Potentials.- 17.A Basic Results.- 17.B Spectral Data for Periodic Potentials of Increasing Period.- 17.C Proof of the Main Theorems.- 18 Unbounded Quasiperiodic Potentials.- 18.A General Results and the Integrated Density of States.- 18.B The Case of Strongly Incommensurate Frequencies.- 18.C The One-Dimensional Case.- 18.D The Schrodinger Operator with a Nonlocal Quasiperiodic Potential.- Problems.- Appendix A: Nevanlinna Functions.- Appendix B: Distribution of Eigenvalues of Large Random Matrices.- List of Symbols.

Spectral properties of disordered systems in the one-body approximation

The paper considers the Schrödinger equation for a single particle and its discrete analogues. Assuming that the coefficients of these equations are homogeneous and ergodic random fields, it is proved that the spectra of corresponding random operators and their point spectra are dense with probability 1 and that in the one-dimensional case they have no absolutely continuous component. Rather wide sufficient conditions of exponential growth of the Cauchy solutions of the one-dimensional equations considered are found.

On the spectrum of random matrices

A study is made of the dis tr ibut ion of eigenvalues in a ce r ta in ensemble of random par t i c les that contains as a special case the ensemble used by Wlgner to give a s ta t i s t ica l descr ip t ion of the energy levels of heavy nuclei, tt is shown that the dis t r ibut ion function of the e lgenvalues divided by the fac tor N {the o r d e r o~ the mat r i ces ) becomes nonrandom In the limit N ~ ~ and can be found by solving a definite functional equation.

Universality of the Local Eigenvalue Statistics for a Class of Unitary Invariant Random Matrix Ensembles

This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n−1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.

A New Approach for Mutual Information Analysis of Large Dimensional Multi-Antenna Channels

This paper adresses the behavior of the mutual information of correlated multiple-input multiple-output (MIMO) Rayleigh channels when the numbers of transmit and receive antennas converge to +infin at the same rate. Using a new and simple approach based on Poincare-Nash inequality and on an integration by parts formula, it is rigorously established that the mutual information when properly centered and rescaled converges to a standard Gaussian random variable. Simple expressions for the centering and scaling parameters are provided. These results confirm previous evaluations based on the powerful but nonrigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community.

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.

ON THE STATISTICAL MECHANICS APPROACH IN THE RANDOM MATRIX THEORY: INTEGRATED DENSITY OF STATES

We consider the ensemble of random symmetricn×n matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn→∞ and that this limiting distribution is the solution of a certain self-consistent equation.

Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model

AbstractWe prove that ifĤNis the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity

Spectral and Probabilistic Aspects of Matrix Models

\bar q_N = N^{ - 1} \sum \left\langle {S_l } \right\rangle _H^2