Entanglement of Collectively Interacting Harmonic Chains

We study the ground-state entanglement of one-dimensional harmonic chains that are coupled to each other by a collective interaction as realized e.g. in an anisotropic ion crystal. Due to the collective type of coupling, where each chain interacts with every other one in the same way,the total system shows critical behavior in the direction orthogonal to the chains while the isolated harmonic chains can be gapped and non-critical. We derive lower and most importantly upper bounds for the entanglement,quantified by the von Neumann entropy, between a compact block of oscillators and its environment. For sufficiently large size of the subsystems the bounds coincide and show that the area law for entanglement is violated by a logarithmic correction.Comment: 5 pages, 1 figur

On a certain class of semigroups of operators

We define an interesting class of semigroups of operators in Banach spaces, namely, the randomly generated semigroups. This class contains as a remarkable subclass a special type of quantum dynamical semigroups introduced by Kossakowski in the early 1970s. Each randomly generated semigroup is associated, in a natural way, with a pair formed by a representation or an antirepresentation of a locally compact group in a Banach space and by a convolution semigroup of probability measures on this group. Examples of randomly generated semigroups having important applications in physics are briefly illustrated.Comment: 11 page

Form factor expansion of the row and diagonal correlation functions of the two dimensional Ising model

We derive and prove exponential and form factor expansions of the row correlation function and the diagonal correlation function of the two dimensional Ising model

Correlations in the Ising antiferromagnet on the anisotropic kagome lattice

We study the correlation function of middle spins, i. e. of spins on intermediate sites between two adjacent parallel lattice axes, of the spatially anisotropic Ising antiferromagnet on the kagome lattice. It is given rigorously by a Toeplitz determinant. The large-distance behaviour of this correlation function is obtained by analytic methods. For shorter distances we evaluate the Toeplitz determinant numerically. The correlation function is found to vanish exactly on a line J_d(T) in the T-J (temperature vs. coupling constant) phase diagram. This disorder line divides the phase diagram into two regions. For J less than J_d(T) the correlations display the features of an unfrustrated two-dimensional Ising magnet, whereas for J greater than J_d(T) the correlations between the middle spins are seen to be strongly influenced by the short-range antiferromagnetic order that prevails among the spins of the adjacent lattice axes. While for J less than J_d(T) there is a region with ferrimagnetic long-range order, the model remains disordered for J greater than J_d(T) down to T=0.Comment: 26 pages, 9 figures, published versio

Convergence to equilibrium for many particle systems

The goal of this paper is to give a short review of recent results of the authors concerning classical Hamiltonian many particle systems. We hope that these results support the new possible formulation of Boltzmann's ergodicity hypothesis which sounds as follows. For almost all potentials, the minimal contact with external world, through only one particle of $N$, is sufficient for ergodicity. But only if this contact has no memory. Also new results for quantum case are presented

Detection and imaging in strongly backscattering randomly layered media

Abstract. Echoes from small reflectors buried in heavy clutter are weak and difficult to distinguish from the medium backscatter. Detection and imaging with sensor arrays in such media requires filtering out the unwanted backscatter and enhancing the echoes from the reflectors that we wish to locate. We consider a filtering and detection approach based on the singular value decomposition of the local cosine transform of the array response matrix. The algorithm is general and can be used for detection and imaging in heavy clutter, but its analysis depends on the model of the cluttered medium. This paper is concerned with the analysis of the algorithm in finely layered random media. We obtain a detailed characterization of the singular values of the transformed array response matrix and justify the systematic approach of the filtering algorithm for detecting and refining the time windows that contain the echoes that are useful in imaging

Statistical M-Estimation and Consistency in Large Deformable Models for Image Warping

The problem of defining appropriate distances between shapes or images and modeling the variability of natural images by group transformations is at the heart of modern image analysis. A current trend is the study of probabilistic and statistical aspects of deformation models, and the development of consistent statistical procedure for the estimation of template images. In this paper, we consider a set of images randomly warped from a mean template which has to be recovered. For this, we define an appropriate statistical parametric model to generate random diffeomorphic deformations in two-dimensions. Then, we focus on the problem of estimating the mean pattern when the images are observed with noise. This problem is challenging both from a theoretical and a practical point of view. M-estimation theory enables us to build an estimator defined as a minimizer of a well-tailored empirical criterion. We prove the convergence of this estimator and propose a gradient descent algorithm to compute this M-estimator in practice. Simulations of template extraction and an application to image clustering and classification are also provided

Spectra of Empirical Auto-Covariance Matrices

We compute spectra of sample auto-covariance matrices of second order stationary stochastic processes. We look at a limit in which both the matrix dimension $N$ and the sample size $M$ used to define empirical averages diverge, with their ratio $\alpha=N/M$ kept fixed. We find a remarkable scaling relation which expresses the spectral density $\rho(\lambda)$ of sample auto-covariance matrices for processes with dynamical correlations as a continuous superposition of appropriately rescaled copies of the spectral density $\rho^{(0)}_\alpha(\lambda)$ for a sequence of uncorrelated random variables. The rescaling factors are given by the Fourier transform $\hat C(q)$ of the auto-covariance function of the stochastic process. We also obtain a closed-form approximation for the scaling function $\rho^{(0)}_\alpha(\lambda)$. This depends on the shape parameter $\alpha$, but is otherwise universal: it is independent of the details of the underlying random variables, provided only they have finite variance. Our results are corroborated by numerical simulations using auto-regressive processes.Comment: 4 pages, 2 figure

Universal correlations of trapped one-dimensional impenetrable bosons

We calculate the asymptotic behaviour of the one body density matrix of one-dimensional impenetrable bosons in finite size geometries. Our approach is based on a modification of the Replica Method from the theory of disordered systems. We obtain explicit expressions for oscillating terms, similar to fermionic Friedel oscillations. These terms are universal and originate from the strong short-range correlations between bosons in one dimension.Comment: 18 pages, 3 figures. Published versio

A Closest Point Proposal for MCMC-based Probabilistic Surface Registration

We propose to view non-rigid surface registration as a probabilistic inference problem. Given a target surface, we estimate the posterior distribution of surface registrations. We demonstrate how the posterior distribution can be used to build shape models that generalize better and show how to visualize the uncertainty in the established correspondence. Furthermore, in a reconstruction task, we show how to estimate the posterior distribution of missing data without assuming a fixed point-to-point correspondence. We introduce the closest-point proposal for the Metropolis-Hastings algorithm. Our proposal overcomes the limitation of slow convergence compared to a random-walk strategy. As the algorithm decouples inference from modeling the posterior using a propose-and-verify scheme, we show how to choose different distance measures for the likelihood model. All presented results are fully reproducible using publicly available data and our open-source implementation of the registration framework