On the transformation from random Schrödinger Hamiltonians to random Hamiltonian matrices

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.

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.

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 define a new diffusive matrix model converging toward the β-Dyson Brownian motion for all β is an element of [0,2] that provides an explicit construction of beta ensembles of random matrices that is invariant under the orthogonal or unitary group. For small values of β, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semicircle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semicircle.

The spectrum edge of random matrix ensembles

22 Moments of the characteristic polynomial in the three ensembles of random matrices

The ensemble exp [−γ Tr (H − H0)2], where the eigenvalues λn of H0 are given by some distribution f(λ), is investigated. In particular, the limits of large and small γ for the orthogonal case are considered. Formal expressions are obtained for various distributions of the eigenvalues in the two limits. The approximation developed in the large-γ limit is also applied to the thermodynamics of an incompletely specified system. Further, it is shown that this approximation is easily extended to include the unitary and symplectic ensembles.

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.

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 use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d=2 (finitely connected XY spins with random chiral interactions) and for d=3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.

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

We consider Markov chains with random transition probabilities which, moreover, fluctuate randomly with time. We describe such a system by a product of stochastic matrices, , with the factors Mi drawn independently from an ensemble of random Markov matrices, whose columns are independent Dirichlet random variables. The statistical properties of the columns of , its largest eigenvalue and its spectrum are obtained exactly for N = 2 and numerically investigated for general N. For large t, the columns are Dirichlet-distributed, however the distribution is different from the initial one. As for the spectrum, we find that the eigenvalues converge to zero exponentially fast and investigate the statistics of the largest stability and Lyapunov exponents, which are approximated by Gamma distributions. We also observe a concentration of the spectrum on the real line for large t.

The correspondence between statistical properties of decaying states and fluctuations in resonance scattering is studied in a statistical model with one open channel. The model is described by an ensemble of random non-Hermitian matrices. The dependence of the correlation length on the coupling parameter both for the S matrix and the cross section is studied numerically. We show that maximal correlations in the scattering arise for a certain value of the coupling to the continuum, reflecting a specific change in the internal motion of intermediate decaying systems. Also, the Fourier transform of the two-point correlation function of the S matrix is analyzed both analytically and numerically. The self-averaging nature of this function is explicitly demonstrated.

Dyson's two-dimensional Coulomb gas on the unit circle with inverse temperature beta is investigated with the help of a Metropolis algorithm. Theoretical predictions for energy and specific heat are verified. The connection with the theory of ensembles of random unitary matrices with orthogonal, unitary or symplectic symmetry corresponding to beta =1, 2, 4 is investigated in detail. New approximation formulae for the spacing distribution and the so-called delta statistics are proposed and found to be useful, not only in the three cases mentioned but also for all beta between 0 and 4.