Finitely connected vector spin systems with random matrix interactions

In this article we consider the effects on ensembles of random Hamiltonian matrices when certain restrictions are imposed on the potentials of the corresponding Schrödinger Hamiltonians. In particular, we investigate the validity of the assumption which is usually made when ensembles of random matrices are used to predict statistical properties of energy levels of complex systems, namely that the Hamiltonian matrix elements are statistically independent.

Random orthogonal matrices are used to randomize integration methods for n-dimensional integrals over spherically symmetric integration regions. Currently available methods for the generation of random orthogonal matrices are reviewed, and some methods for the generation of quasi-random orthogonal matrices are proposed. These methods all have O(n 3) time complexity. Some new methods to generate both random and quasi-random orthogonal matrices will be described and analyzed. The new methods use products of butterfly matrices, and have time complexity O(log(n)n 2). The use of these methods will be illustrated with results from the numerical computation of high-dimensional integrals from a computational finance application.

Stochastic properties of many-body systems

A Fourier analysis of parametric level dynamics for random matrices periodically depending on a phase is developed. We demonstrate both theoretically and numerically that under very general conditions the correlation C(cphi) of level velocities is singular at cphi=0 for any symmetry class; the singularity is revealed by algebraic tails in Fourier transforms, and is milder, the stronger the level repulsion in the chosen ensemble. The singularity is strictly connected with the divergence of the second moments of level derivatives of appropriate order, and its type is specified to leading terms for Gaussian, stationary ensembles of orthogonal (GOE), unitary (GUE), and symplectic (GSE) types, and for the Gaussian ensemble of periodic banded random matrices, in which a breaking of symmetry occurs. In the latter case, we examine the behavior of correlations in the diffusive regime and in the localized one as well, finding a singularity like that of pure GUE cases. In all the considered ensembles we study the statistics of the Fourier coefficients of eigenvalues, which are Gaussian distributed for low harmonics, but not for high ones, and the distribution of kinetic energies.

Complex systems are typically represented by large ensembles of observations. Correlation matrices provide an efficient formal framework to extract information from such multivariate ensembles and identify in a quantifiable way patterns of activity that are reproducible with statistically significant frequency compared to a reference chance probability, usually provided by random matrices as fundamental reference. The character of the problem and especially the symmetries involved must guide the choice of random matrices to be used for the definition of a baseline reference. For standard correlation matrices this is the Wishart ensemble of symmetric random matrices. The real world complexity however often shows asymmetric information flows and therefore more general correlation matrices are required to adequately capture the asymmetry. Here we first summarize the relevant theoretical concepts. We then present some examples of human brain activity where asymmetric time-lagged correlations are evident and hence highlight the need for further theoretical developments.

The quantum chaos conjecture applied to a finite dimensional quantum system implies that such a system has eigenstates that show similar statistical properties as the column vectors of random orthogonal or unitary matrices. Here, we consider the different probabilities for obtaining a specific outcome in a projective measurement, provided the system is in one of its eigenstates. We then give analytic expressions for the joint probability density for these probabilities, with respect to the ensemble of random matrices. In the case of the unitary group, our results can be applied, also, to the phenomenon of universal conductance fluctuations, where the same mathematical quantities describe partial conductances in a two-terminal mesoscopic scattering problem with a finite number of modes in each terminal.

In this paper we apply the REFORM method[3, 4] to the deduction of the system of canonical equations for normalized spectral functions of the matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is random matrix from class C 12 [26] or from the following class C 13 of distributions of random matrices: where are independent random real matrices whose entries are independent for every n, and for a certain δ > 0 the Lyapunov condition is fulfilled: This problem has been considered in some publications for matrices A n + U n , where Unitary matrix U n from the class C 1 [26] is distributed by Haar measure, at the “ad hoc” level, without a strong proof, on the basis of heuristic calculations. Therefore, the behavior of limit n.s.f. of the sum of a random Unitary matrix U n and a nonrandom matrix A n has not been discovered. Many conclusions of various kinds have been presented in the literature (see [13–27]) for random matrices A n + Ξ n B n , where Ξ n is the random matrix with independent entries, concerning, for example, the effectiveness of the REFORM method and the role of the martingale difference representation for the resolvent of random matrices. We give the formulation of the Circular Law for random matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is from the so called Class C 12 of random Unitary matrices (see[26]). In this case the Circular law means that the support of accompanying spectral density p ( x, y ) of eigenvalues of A n + B n U n ( ε ) C n looks like several drops of mercury on a table, and inside of some drops it is possible that some dry circles appear. We call this support of limit density p ( x, y ) the Mercury support . If the distances between the centers of these drops are large enough we have several separate almost circular drops as for corresponding description of limit spectral density for the Hermitian random matrices [28, 29]. According to the distances between the centers of these drops, these drops might not touch each other or can merge creating fanciful shapes represented at the end of the paper, Figures 1–16. The analogy with the Circular Law is the following: for a simple Unitary matrix U n from class C 1 all its eigenvalues are distributed asymptotically uniformly on the circumference and, evidently, this limit distribution does not coincide with the uniform distribution on the circle. But if we will add to Unitary matrix U n any diagonal nonrandom matrix A n = ( δ ij a j ) then if the distances between diagonal entries a j are large enough we have almost the same picture of limit distribution of eigenvalues of U n + A n if Unitary matrix U n would be equal to the random matrix Ξ n with random independent entries satisfying the Global Circular Law (see Part I of this paper). The rough explanation of this phenomenon is the following: the expected scalar products of the vector row or vector column of matrix Ξ n have the order n −1 . Therefore, the matrix Ξ n looks like orthogonal random matrix. In this paper we use for matrices A n + B n U n ( ε ) C n the triply regularized V -transform[2, 5],( VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Randomly Independent Arrays ) where α > 0, ε ≠= 0, γ ≠= 0 are regularization parameters , τ = t + is is a complex number and µ n { x,G n ( t, s, γ, ε )} is the normalized spectral function of G -matrix: Such a V -transformation was used for the first time in 1982–1985 in [2,3,5]. Canonical equations for the limit distribution function of normalized spectral functions of some matrices A n + B n U n and the following V - domain G δ , δ > 0(see [27, pp.142, 173]) of the pseudodistribution of eigenvalues of matrices A n + B n U n are found: where are eigenvalues of the matrix.

We study the eigenvalue distribution of dilute N×N random matrices HN that in the pure (undiluted) case describe the Hopfield model. We prove that for the fixed dilution parameter a the normalized counting function (NCF) of HN converges as N→∞ to a unique σa(λ). We find the moments of this distribution explicitly, analyze the 1/a correction, and study the asymptotic properties of σa(λ) for large |λ|. We prove that σa(λ) converges as a →∞ to the Wigner semicircle distribution (SCD). We show that the SCD is the limit of the NCF of other ensembles of dilute random matrices. This could be regarded as evidence of stability of the SCD to dilution, or more generally, to random modulations of large random matrices.

We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, W=XX(†), where X stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, MP(⊠s), which for an integer s yield Fuss-Catalan distributions corresponding to a product of s-independent square random matrices, X=X(1)⋯X(s). New formulas for the level densities are derived for s=3 and s=1/3. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions, is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.

Distributions of the resilience of transport networks are studied numerically, in particular the large-deviation tails. Thus, not only typical quantities like average or variance but the distributions over the (almost) full support can be studied. For a proof of principle, a simple transport model based on the edge-betweenness and three abstract yet widely studied random network ensembles are considered here: Erdős-Rényi random networks with finite connectivity, small world networks and spatial networks embedded in a two-dimensional plane. Using specific numerical large-deviation techniques, probability densities as small as 10-80 are obtained here. This allows to study typical but also the most and the least resilient networks. The resulting distributions fulfill the mathematical large-deviation principle, i.e., can be well described by rate functions in the thermodynamic limit. The analysis of the limiting rate function reveals that the resilience follows an exponential distribution almost everywhere. An analysis of the structure of the network shows that the most-resilient networks can be obtained, as a rule of thumb, by minimizing the diameter of a network. Also, trivially, by including more links a network can typically be made more resilient. On the other hand, the least-resilient networks are very rare and characterized by one (or few) small core(s) to which all other nodes are connected. In total, the spatial network ensemble turns out to be most suitable for obtaining and studying resilience of real mostly finite-dimensional networks. Studying this ensemble in combination with the presented large-deviation approach for more realistic, in particular dynamic transport networks appears to be very promising.

We investigate the probability distribution ${\mathcal{P}}_{N}(r)$ of the radius of gyration $r$ of a polymer chain of size $N$ with excluded-volume interactions at infinite temperature. This function shows the geometric contribution to the tricritical coil-globule transition of self-avoiding walks; it indicates that the relevant order parameter $t$ of the transition is a power of the density $\ensuremath{\rho}{=Nr}^{\ensuremath{-}d}.$ The theoretical form of the distribution ${P}_{N}(t)$ of this order parameter is deduced from scaling arguments, and supported by numerical simulations. Intending to probe the contribution of the different subsets of conformations, namely, globule, coil and stretch, we supplement ${P}_{N}(t)$ with a formal Boltzmann factor; this model undergoes a tricritical coil-globule transition which is solved exactly. We show a nontrivial finite-size scaling for ${P}_{N}(t)$ and analyze its convergence toward the thermodynamic limit. Due to the presence in ${P}_{N}(t)$ of a diverging factor ${t}^{c}$ with $cl\ensuremath{-}1,$ this convergence happens to be tragically slow. As a result, the scaling behavior observed in numerical simulations is qualitatively different from its thermodynamic limit, and we relate the critical exponents of the geometric transition in the thermodynamic limit and the effective exponents observed at finite size.

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature β. For 2×2 matrices with Gaussian distribution we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective β_{eff}(β) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of β. It reproduces the 2D Poisson distribution at β=0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values β=1.4 and β=2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.

We consider an ensemble of Wigner symmetric random matrices An={aij}, i,j=1, . . . ,n with matrix elements aij, being i.i.d. symmetrically distributed random variables Open image in new window We assume that Open image in new window and that

We consider an ensemble of nxn real symmetric random matrices A whose entries\nare determined by independent identically distributed random variables that\nhave symmetric probability distribution. Assuming that the moment 12+2delta of\nthese random variables exists, we prove that the probability distribution of\nthe spectral norm of A rescaled to n^{-2/3} is bounded by a universal\nexpression. The proof is based on the completed and modified version of the\napproach proposed and developed by Ya. Sinai and A. Soshnikov to study high\nmoments of Wigner random matrices.\n

Some well-known real-number codes are DFT codes. Since these codes are cyclic, they can be used to correct erasures (errors at known positions) and detect errors, using the locator polynomial via the syndrome, with efficient algorithms. The stability of such codes are, however, very poor for burst error patterns. In such conditions, the stability of the system of equations to be solved is very poor. This amplifies the rounding errors inherent to the real number field. In order to improve the stability of real-number error-correcting codes, other types of coding matrices were considered, namely random orthogonal matrices. These type of codes have proven to be very stable, when compared to DFT codes. However, the problem of detecting errors (when the positions of these errors are not known) with random codes was not addressed. Such codes do not possess any specific structure which could be exploited to create an efficient algorithm. In this paper, we present an efficient method to locate errors with codes based on random orthogonal matrices. Index Terms — Error correction, random matrices, real number codes, sparse solutions