Temperature Correlation Function in the Absolutely Anisotropic XXZ Heisenberg Magnet

For the one-dimensional XXZ Heisenberg chain of spin 1/2 at = –, we calculate the two-time temperature correlation function of the third components of local spins. For the correlation function in the thermodynamic limit, we obtain the expression in terms of Fredholm determinants for linear integral operators.     

A New Algorithm for Simulating a Correlation Matrix Based on Parameter Expansion and Reparameterization

The correlation matrix (denoted by R) plays an important role in many statistical models. Unfortunately, sampling the correlation matrix in Markov chain Monte Carlo (MCMC) algorithms can be problematic. In addition to the positive definite constraint of covariance matrices, correlation matrices have diagonal elements fixed at one. In this article, we propose an efficient two-stage parameter expanded reparameterization and Metropolis-Hastings (PX-RPMH) algorithm for simulating R. Using this algorithm, we draw all elements of R simultaneously by first drawing a covariance matrix from an inverse Wishart distribution, and then translating it back to a correlation matrix through a reduction function and accepting it based on a Metropolis-Hastings acceptance probability. This algorithm is illustrated using multivariate probit (MVP) models and multivariate regression (MVR) models with a common correlation matrix across groups. Via both a simulation study and a real data example, the performance of the PX-RPMH algorithm is compared with those of other common algorithms. The results show that the PX-RPMH algorithm is more efficient than other methods for sampling a correlation matrix.     

Hermitean Matrix Ensembles and Orthogonal Polynomials

In this article, we investigate orthogonal polynomials associated with complex Hermitean matrix ensembles using the combination of the methods of Coulomb fluid (or potential theory), chain sequences, and Birkhoff–Trjitzinsky theory. We give a general formula for the largest eigenvalue of the N×N Jacobi matrices (which is equivalent to estimating the largest zero of a sequence of orthogonal polynomials) and the two-level correlation function for the α ensembles (α>0) introduced previously for α>1. In the case of 0<α<1, we give a natural representation for the weight function that is a special case of the general Nevanlinna parametrization. We also discuss Hermitean matrix ensembles associated with general indeterminate moment problems.     

Time-Dependent Correlators of Local Spins of the One-Dimensional XY Heisenberg Chain

Time-dependent temperature correlators of the anisotropic Heisenberg XY chain are calculated by the integration techniques with respect to Grassmann variables. For a chain of length M, the correlators are represented as the determinants of 2M × 2M matrices. In the thermodynamic limit, the correlation functions are expressed in terms of the Fredholm determinants of linear integral operators with matrix kernels. Bibliography: 32 titles.     

Integral Representations for Correlation Functions of the XXZ Heisenberg Chain

We consider new integral representations for two-point correlation functions of local spins 1/2 in the XXZ Heisenberg chain.     

Detection of gross errors by the chauvenet test for observations connected in a homogeneous markov chain

The Chauvenet rule for detection of gross errors is extended for the case of a stationary Gaussian Markov chain with unknown correlation. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 137–142, 1993. Translated by L. Khalfin.     

Extremal dependence measure and extremogram: the regularly varying case

The dependence of large values in a stochastic process is an important topic in risk, insurance and finance. The idea of risk contagion is based on the idea of large value dependence. The Gaussian copula notoriously fails to capture this phenomenon. Two notions in a process or vector context which summarize extremal dependence in a function comparable to a correlation function are the extremal dependence measure (EDM) and the extremogram. We review these ideas and compare the two tools and end with a central limit theorem for a natural estimator of the EDM which allows drawing confidence bands comparable to those provided by Bartlett’s formula in a classical context of sample correlation functions.     

On correlation functions of the XY model

New determinant representations are obtained for the simplest correlation functions of the Heisenberg XY spin chain. Bibliography: 8 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 178–191.     

Calculation of the asymptotics of the two-point correlation function for one-dimensional Bose gas

We consider a quantum field-theoretical model which describes spatially-nonhomogeneous, one-dimensional, repulsive Bose gas in an external harmonic potential. The two-point correlation function is calculated in the framework of functional integration. The corresponding functional integrals are estimated by means of stationary phase approximation. Asymptotic estimates are obtained in the limit as the temperature tends to zero while the volume occupied by the quasi-condensate increases. A power-law behavior is established for the correlation function in this limit. It is shown that the power-law behavior is governed by the critical exponent depending on spatial arguments. Bibliography 32 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 56–74.     

Orientations of chain groups

We prove generalizations to chain groups, of Minty's Arc Colouring Lemma and its extension, the well-known Farkas Lemma. In these the orientation of the edges is replaced by an arbitrary chain.A function on a chain groupN isrepresentable if there exists a chainR such that (X)=R·X for allXN. Anorientation is a chain with values ±1. We prove that for a regular chain group a linear function that is representable by an orientation for each chainXN locally, is representable by an orientation globally.     

On the distribution of imaginary parts of zeros of the Riemann zeta function,II

We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Montgomery’s pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. The first author is supported by National Science Foundation Grant DMS-0555367. The second author is partially supported by the National Science Foundation and the American Institute of Mathematics (AIM). The third author is supported by National Science Foundation Grant DMS-0456615.     

Approximations to the distribution of the sample correlation matrix

In this article, multivariate density expansions for the sample correlation matrix R are derived. The density of R is expressed through multivariate normal and through Wishart distributions. Also, an asymptotic expansion of the characteristic function of R is derived and the main terms of the first three cumulants of R are obtained in matrix form. These results make it possible to obtain asymptotic density expansions of multivariate functions of R in a direct way.     

A special value of the spectral zeta function of the non-commutative harmonic oscillators

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function. The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) 15340005 from the Ministry of Education, Culture, Sports, Science and Technology.     

Computation of moment functions for parabolic PDEs with random parameters

This paper deals with different approaches for the computation of stochastic characteristics (such as mean, variance, correlation function, …) of the solution to parabolic PDEs with random parameters, in particular with a random Neumann boundary conditions and a random initial condition. Thereby Finite-Element discretisation of the spatial variables is used for the construction of pathwise solutions. The random influences are assumed to be ε -correlated (cp. [4]) that means the correlation functions vanish if the difference of the arguments exceeds a given correlation length ε . So an asymptotic expansion of higher order with respect to the correlation length for the correlation function of the approximative solution is given. A second possibility is the so called explicit calculation. Examples which compare the different methods can be found in [2]. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Theory of KM2O-Langevin Equations for Stationary Flows (2): Construction Theorem

The aim of the present paper is to construct the KM₂O-Langevin matrix LM([X,Y]) directly from the correlation matrix function R by using (DDT), (FDT) and (PAC) as an algorithm.     

Spin chain related to the quantum superalgebra slq(1¦1)

We consider an integrable system with R-matrix related to the algebra sl _q(1 | 1). The Hamiltonian of the system is constructed, and its spectrum is found by means of the algebraic Bethe ansatz. The symmetry algebra of the chain is written out. The partition function of the model on the lattice with domain wall boundary conditions is calculated. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 146–162.     

A test for a hypothesis on the correlation function of Gaussian random processes

A test is proposed for a hypothesis on the correlation function of general Gaussian random processes. The test is based on theorems on estimates of the distribution of the supremum of sample estimators of correlation functions of Gaussian processes. For a wide class of stationary processes formulas are given that allow the test to be used immediately. Bibliography: 4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 61–74.     

Ising limit of a Heisenberg XXZ magnet and some temperature correlation functions

We consider the Heisenberg spin-1/2 XXZ magnet in the case where the anisotropy parameter tends to infinity (the so-called Ising limit). We find the temperature correlation function of a ferromagnetic string above the ground state. Our approach to calculating correlation functions is based on expressing the wave function in the considered limit in terms of Schur symmetric functions. We show that the asymptotic amplitude of the above correlation function at low temperatures is proportional to the squared number of strict plane partitions in a box.