Random matrices and controllability of dynamical systems

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

The microscopic spectral correlations of the Dirac operator in Yang-Mills theories coupled to fermions in $2+1$ dimensions can be related to three universality classes of random matrix theory. In the microscopic limit the orthogonal ensemble corresponds to a theory with two colors and fermions in the fundamental representation and the symplectic ensemble corresponds to an arbitrary number of colors and fermions in the adjoint representation. Using the new method of Widom, we derive an expression for the two scalar kernels which through quaternion determinants gives all spectral correlation functions in the Gaussian orthogonal ensemble (GOE) and in the Gaussian symplectic ensemble (GSE) with all fermion masses equal to zero. The result for the GOE is valid for an arbitrary number of fermions while for the GSE we have results for an even number of fermions.

Spectral form factor for chaotic dynamics in a weak magnetic field

Formulas are derived for the average level density of deformed, or transition, Gaussian orthogonal random matrix ensembles. After some general considerations about Gaussian ensembles, we derive formulas for the average level density for (i) the transition from the Gaussian orthogonal ensemble (GOE) to the Poisson ensemble and (ii) the transition from the GOE to m GOEs.

We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erds–Rényi (E–R) random graph is determined by the average degree ⟨k⟩ and p(s) undergoes a dramatic change when ⟨k⟩ is varied around the critical point of the percolation transition, ⟨k⟩ = 1. When ⟨k⟩ ≫ 1, the p(s) is described by the statistics of the Gaussian orthogonal ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at ⟨k⟩ = 1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brody-distribution. Furthermore, below the critical point p(s) can be given with the help of the regularized Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the internet, a word association network and a protein–protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E–R graph.

These lecture notes provide a comprehensive self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics computational mathematics and engineering sciences including data assimilation signal processing combinatorial optimization compressed sensing econometrics and mathematical finance among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics non-commutative algebra geometry multivariate functional and spectral analysis and of course statistics and probability theory. As a result most of the classical topics in random matrix theory are technical and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specificcontext of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory. We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied along with new spectral and trace concentration-type inequalities.

Statistics of the eigenvalues of Tsallis matrices

We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials studied by NM in the case of the GUE. With these polynomials we obtain expressions for the cumulative distribution functions of the largest eigenvalue in the GOE and the GSE. Further, by performing asymptotic analysis of these skew-orthogonal polynomials in the limit of large matrix size, we obtain an alternative derivation of the Tracy-Widom distributions for GOE and GSE. This asymptotic analysis relies on a certain Pfaffian identity, the proof of which employs the characterization of Pfaffians in terms of perfect matchings and link diagrams.

This article discusses the connection between a particular class of growth processes and random matrices. It first provides an overview of growth model, focusing on the TASEP (totally asymmetric simple exclusion process) with parallel updating, before explaining how random matrices appear. It then describes multi-matrix models and line ensembles, noting that for curved initial data the spatial statistics for large time t is identical to the family of largest eigenvalues in a Gaussian Unitary Ensemble (GUE multi-matrix model. It also considers the link between the line ensemble and Brownian motion, and whether this persists on Gaussian Orthogonal Ensemble (GOE) matrices by comparing the line ensembles at fixed position for the flat polynuclear growth model (PNG) and at fixed time for GOE Brownian motions. Finally, it examines (directed) last passage percolation and random tiling in relation to growth models.

Random matrix theory models of electric grid topology

The structure of this survey is the following: at first we repeat in Part I the first 20 years old proof of the strong Elliptic law for random matrices Then in Part II we prove the strong Elliptic law for random matrices Ξ n of the general form, i.e. when their diagonal entries have nonzero expectations, and when we do not require the existence of the probability densities of the entries of random matrices, but instead of Lyapunov condition we require the boundedness of the absolute moments of the entries of the order 4 + δ , δ &gt; 0. In this case the Elliptic law means that the support of the accompanying spectral density of eigenvalues looks like the picture of several galaxies made by telescope. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. We also present the sand clock law density for the sum of diagonal complex matrix and the symmetric random matrix belonging the domain of attraction of the Semicircular law. After this we prove in the Part III the weak Elliptic law, when all entries can have nonzero expectations. At the end of survey we give the proof of the Elliptic law for the Unitary random matrices from the so called Class N 12 of random Unitary matrices and give the over turned stool law density for the sum of diagonal complex matrix A n and where Ξ n belongs to the domain of attraction of the Circular law. We also establish in Part III for Elliptic Law at the first time the convergency rate and the Central Limit Theorem. These statements are based on the VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays. We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions where λ k ( A n Ξ n B n + C n ) are eigenvalues of the matrix A n Ξ n B n + C n , A n , B n , and C n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ n , χ is the indicator function. We emphasize that the spectral theory of Hermitian random matrices is rather profound. For example, in 1975 in [3] V. Girko proved the general stochastic canonical equation for ACE (Asymptotically Constant Entries)-symmetric matrices: Assume that for any n , the random entries of a symmetric matrix are independent and they are asymptotically constant entries (ACE), i.e., for any ε &gt; 0, and τ &gt; 0 is an arbitrary constant, and that, for every 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, where the symbol denotes the weak convergence of distribution when n → ∞, is a nondecreasing function with bounded variation in z and continuous in u and v in the domain 0 ≤ u , v ≤ 1. Then, with probability one, for almost all x , where λ k (Ξ n × n ) are eigenvalues, F ( x ) is a distribution function whose Stieltjes transform satisfies the relation G α ( x , y , t ), as a function of x , is a distribution function satisfying the regularized stochastic canonical equation K 3 [3, 23] at the points x of continuity, is a random real functional whose Laplace transform of one-dimensional distribution is equal to The integrand <jats:inli

Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-D integrable model with the 1/r2 interaction (the Calogero-Sutherland-Moser system) and 2-D quantum gravity. We review the connection of RMT with these areas. We also discuss the method of loop equations for determining correlation functions in RMT, and smoothed global eigenvalue correlators in the two-matrix model for Gaussian orthogonal, unitary and symplectic ensembles.

We study the spectral properties of and spectral crossovers between different random matrix ensembles [Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE)] in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term, competing with the usual isotropic and time-reversal invariant Heisenberg term. We have investigated these crossovers in the context of the level-spacing distribution and the level-spacing ratio distribution. We use random matrix theory (RMT) analytical results to fit the observed Poissonian-to-GOE and GOE-to-GUE crossovers, and examine the relationship between the RMT crossover parameter $\ensuremath{\lambda}$ and scaled physical parameters of the spin-chain systems in terms of a scaling exponent. We find that the crossover behavior exhibits universality, in the sense that, it becomes independent of lattice size in the large Hamiltonian matrix dimension limit. Moreover, the scaling exponent obtained from such a finite size scaling analysis, seems to be quite robust and independent of the type of crossover considered or the specific spectral correlation measure used.

We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlev #x27;e II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. We also give sketches of the results for the limiting distribution of the largest eigenvalue in the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble (GSE). This work we did some years ago in a more general setting. These notes, therefore, are not meant for experts in the field.

If V(r) is homogeneous in the position coordinates, then the Schr\"odinger equation with H(t)=T+\ensuremath{\alpha}(t)V(r) is separable in time if \ensuremath{\alpha}(t) is properly chosen. This fact makes the time evolution of such systems much easier to compute than it would otherwise be. Examples of such systems are the hydrogen atom with time-varying nuclear charge, the harmonic oscillator with a changing spring constant, and the infinite square well with moving walls. If \ensuremath{\alpha}(t) is made periodic by concatenating ``properly chosen'' segments, then a unitary operator U can be defined which evolves the system through one period. If the wave function is expressed as an N-term truncated expansion of eigenfunctions of H(t), then U is an N\ifmmode\times\else\texttimes\fi{}N matrix which can be computed straightforwardly. The properties of U are studied by examining the statistical properties of its eigenvectors and eigenvalues, whose phases are the quasienergies. Numerical examples are given for the Fermi-Ulam cosmic accelerator (FUCA), which is a particle in a one-dimensional box with a moving wall, with various values for relevant parameters, which can lead to Poisson, Gaussian orthogonal ensemble, Gaussian unitary ensemble, or other quasienergy statistics, corresponding to inte- grable, chaotic, and non-time-reversible chaotic situations. Poincar\'e sections for the classical FUCA are also displayed. They show regimes of regular and of chaotic motion, whose correspondence with the statistics of U is sought with limited success.