Showing papers on "Randomness published in 1989"

An invariance principle for reversible Markov processes. Applications to random motions in random environments

Abstract: We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.

Thermodynamic cost of computation, algorithmic complexity and the information metric

Abstract: Algorithmic complexity is discussed as a computational counterpart to the second law of thermodynamics. It is shown that algorithmic complexity, which is a measure of randomness, sets limits on the thermodynamic cost of computations and casts a new light on the limitations of Maxwell's demon. Algorithmic complexity can also be used to define distance between binary strings.

Random-field mechanism in random-bond multicritical systems.

Abstract: It is argued on general grounds that bond randomness drastically alters multicritical phase diagrams via a random-field mechanism. For example, tricritical points and critical end points are entirely eliminated (d\ensuremath{\le}2) or depressed in temperature (dg2). These predictions are confirmed by a renormalization-group calculation. Another consequence of this phenomenon is that, under bond randomness, the phase transitions of q-state Potts models are second order for all q at dimensionality d\ensuremath{\le}2.

Numerical study of symmetry effects on localization in two dimensions.

Abstract: Effects of symmetry on localization on two-dimensional square lattices are studied numerically. The inverse localization length is determined by the system-size dependence of the Thouless number in magnetic fields or in the presence of strong spin-orbit interactions. A finite-size-scaling method is also applied to the case of spin-orbit interactions. Extended states, present in each Landau level in strong magnetic fields or in the case of small randomness, are found to merge together with increasing randomness and disappear beyond a certain critical randomness. In weak magnetic fields, the field tends to reduce the localization near the band center, while the localization is enhanced in the band-tail region. Spin-orbit interactions cause effects similar to, but much stronger than, that due to a weak magnetic field. The critical randomness and exponent for a metal-insulator transition are determined in the presence of strong spin-orbit interactions.

The Random Utility Hypothesis and Inference in Demand Systems

Abstract: In this paper, the authors examine the consequences of adopting the random utility hypothesis as an approach for randomizing a system of demand equations. Random utility models are appealing since they allow the usual assumption of deterministic utility-maximizing behavior by each consumer to coexist with the apparent randomness across individuals that is exhibited by data. Their results show that the use of random utility models implies that the disturbances of the demand equations may not be homoskedastic, but must be functions of prices and/or income. Copyright 1989 by The Econometric Society.

Mean first-passage time of random walks on a random lattice

Abstract: The mean first-passage time (MFPT) of random walks on one-dimensional disordered lattice segments is considered. Disorder is modeled by prescribing random transition and sojourn probabilities to the lattice sites. We consider several models of disorder: models with symmetric and asymmetric random transition probabilities, random sojourn probabilities, and models with bond randomness. For these models we present exact results on the MFPT and disorder-averaged MFPT. We do not find any anomalous dependence of the disorder-averaged MFPT on the size of the lattice segment. The distribution of the MFPT resulting from the disorder is Gaussian for the models with symmetric and site symmetric transition probabilities and non-Gaussian for models with asymmetric transition probabilities.

Polymers on Disordered Hierarchical Lattices: A Nonlinear Combination of Random Variables

Abstract: The problem of directed polymers on disordered hierarchical and hypercubic lattices is considered For the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way We present the results of numerical simulations of two hierarchical lattices, finding evidence of a phase transition in one case For a limiting case we extend the perturbation theory developed by Derrida and Griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties (overlaps) of the model In this limit we obtain an interpolation formula, allowing one to obtain the noninteger moments of the partition function from the integer moments We obtain bounds for the transition temperature for hierarchical and hypercubic lattices, and some similarities between the problem on the two different types of lattice are discussed

Testing for a Random Walk: A Simulation Experiment of Power When the Sampling Interval is Varied

Abstract: This paper analyzes the power of various tests for the random walk hypothesis against AR(1) alternatives when the sampling interval is allowed to vary. The null and alternative hypotheses are set in terms of the parameters of a continuous time model. The discrete time representations are derived and it is shown how they depend on the sampling interval. The power is simulated for a grid of values of the number of observations and the span of the data available (hence for various sampling intervals). Various test statistics are considered among the following classes: (a) test for a unit root on the original series and (b) tests for randomness in the differenced series. Among class (b), we consider both parametric and nonparametric tests, the latter including tests based on the rank of the first-differenced series. The paper therefore not only provides information as to the relative power of these tests but also about their properties when the sampling interval varies. This work is an extension of Perron (1987) and Shiller and Perron (1985).

Quasiperiodicity and randomness in tilings of the plane

Abstract: We define new tilings of the plane with Robinson triangles, by means of generalized inflation rules, and study their Fourier spectrum. Penrose's matching rules are not obeyed; hence the tilings exhibit new local environments, such as three different bond lengths, as well as new patterns at all length scales. Several kinds of such generalized tilings are considered. A large class of deterministic tilings, including chiral tilings, is strictly quasiperiodic, with a tenfold rotationally symmetric Fourier spectrum. Random tilings, either locally (with extensive entropy) or globally random (without extensive entropy), exhibit a mixed (discrete+continuous) diffraction spectrum, implying a partial perfect long-range order.

Psychological conceptions of randomness

Abstract: This article presents a critique of the concept of randomness as it occurs in the psychological literature. The first section of our article outlines the significance of a concept of randomness to the process of induction; we need to distinguish random and non-random events in order to perceive lawful regularities and formulate theories concerning events in the world. Next we evaluate the psychological research that has suggested that human concepts of randomness are not normative. We argue that, because the tasks set to experimental subjects are logically problematic, observed biases may be an artifact of the experimental situation and that even if such biases do generalise they may not have pejorative implications for induction in the real world. Thirdly we investigate the statistical methodology utilised in tests for randomness and find it riddled with paradox. In a fourth section we find various branches of scientific endeavour that are stymied by the problems posed by randomness. Finally we briefly mention the social significance of randomness and conclude by arguing that such a fundamental concept merits and requires more serious considerations.

Linear elasticity of planar delaunay networks: Random field characterization of effective moduli

Abstract: A study is conducted of the influence of microscale geometric and physical randomness on effective moduli of a continuum approximation of disordered microstructures. A particular class of microstructures investigated is that of planar Delaunay networks made up of linear elastic rods connected by joints. Three types of networks are considered: Delaunay networks with random geometry and random spring constants, modified Delaunay networks with random geometry and random spring constants, and regular triangular networks with random spring constants. Using a structural mechanics method, a numerical study is conducted of the first and second order characteristics of random fields of effective moduli. In view of duality of the Delaunay triangulations to the Voronoi tessellations, these results provide the basis for development of analytical models of various heterogeneous solids, e.g. granular, fibrous.

Small random perturbations of finite and infinite dimensional dynamical systems: unpredictability of exit times

Abstract: We apply previous results on the pathwise exponential loss of memory of the initial condition for stochastic differential equations with small diffusion to the problem of the asymptotic distribution of the first exit times from an attracted domain. We show under general hypotheses that the suitably rescaled exit time converges in the zero-noise limit to an exponential random variable. Then we extend the results to an infinite-dimensional case obtained by adding a small random perturbation to a nonlinear heat equation.

Ergodic Theory, Randomness, and "Chaos"

Abstract: Ergodic theory is the theory of the long-term statistical behavior of dynamical systems. The baker's transformation is an object of ergodic theory that provides a paradigm for the possibility of deterministic chaos. It can now be shown that this connection is more than an analogy and that at some level of abstraction a large number of systems governed by Newton's laws are the same as the baker's transformation. Going to this level of abstraction helps to organize the possible kinds of random behavior. The theory also gives new concrete results. For example, one can show that the same process could be produced by a mechanism governed by Newton's laws or by a mechanism governed by coin tossing. It also gives a statistical analog of structural stability.

Statistical Characteristics of Ocean Acoustic Noise Processes

Abstract: A statistical analysis is given of ambient noise data from several ocean acoustic environments. Included in the analysis are statistical tests for homogeneity and randomness, statistical tests for normality, sample autocorrelation functions, and kernel density estimates of the instantaneous amplitude fluctuations. The test results indicate that a randomness hypothesis may be rejected when Nyquist rate sampling is employed. A randomization procedure is applied to the data in order to create ensembles which pass the tests for randomness and homogeneity. Analysis of these ensembles indicates that a stationary Gaussian assumption is not justified for some ocean environments. The largest deviations from normality occur in the tail regions of the density function and are often attributable to non-stationary characteristics of the data.

Exponential convergence to equilibrium for a class of random-walk models

Abstract: We prove exponential convergence to equilibrium (L 2 geometric ergodicity) for a random walk with inward drift on a sub-Cayley rooted tree. This randomwalk model generalizes a Monte Carlo algorithm for the self-avoiding walk proposed by Berretti and Sokal. If the number of vertices of levelN in the tree grows asC N ~μ N N γ−1 , we prove that the autocorrelation time τ satisfies 〈N〉2 ≲ τ ≲ 〈N〉1+γ

Transport properties of the continuous-time random walk with a long-tailed waiting-time density

Abstract: We derive asymptotic properties of the propagatorp(r,t) of a continuous-time random walk (CTRW) in which the waiting time density has the asymptotic formψ(t)∼Tα/tα+1 whent≫T and 0<α<1. Several cases are considered; the main ones are those that assume that the variance of the displacement in a single step of the walk is finite. Under this assumption we consider both random walks with and without a bias. The principal results of our analysis is that one needs two forms to characterizep(r,t), depending on whetherr is large or small, and that the small-r expansion cannot be characterized by a scaling form, although it is possible to find such a form for larger. Several results can be demonstrated that contrast with the case in which 〈t〉=∫0∞τψ(τ)dτ is finite. One is that the asymptotic behavior ofp(0,t) is dominated by the waiting time at the origin rather than by the dimension. The second difference is that in the presence of a fieldp(r,t) no longer remains symmetric around a moving peak. Rather, it is shown that the peak of this probability always occurs atr=0, and the effect of the field is to break the symmetry that occurs when 〈t〉∞. Finally, we calculate similar properties, although in not such great detail, for the case in which the single-step jump probabilities themselves have an infinite mean.

Existence of Anderson localization of classical waves in a random two-component medium

Abstract: An exact mapping of the classical wave problem to that of electronic motion is utilized together with extensive numerical results to examine the question of the existence of genuine localization (i.e., one occurring when both components have real positive dielectric constants) of classical waves in random binary alloys A/sub 1-//sub x/B/sub x/. We find that scalar waves do exhibit localization. We have also developed a coherent potential approximation which for x<0.2 gives results not that much different from the numerical ones. This result can be easily generalized to electromagnetic fields as well.

Random close packing (RCP) of equal spheres: structure and implications for use as a model porous medium

Abstract: The structure of the Finney Random Close Packing (RCP) of equal spheres has been analysed, together with the influence which such structure exerts over the capillary pressure characteristics of geometrically similar sphere packings. The analysis is centred on the simplicial, or Delaunay cell, which is an irregular tetrahedron with apices defined by four immediate neighbour sphere-centres. In terms of using RCP as a model porous medium, an individual simplicial cell is equivalent to an individual pore. A number of measured pore-size distribution parameters are presented for the Finney packing, from which it is shown from first principles that drainage-imbibition hysteresis is not an intrinsic property of the individual pore. The nature and degree of randomness which characterises the Finney packing is evaluated on two levels. First, by classifying edgelengths as either short or long, seven mutually exclusive cell classes are defined. Using the binomial theorem it is shown that cells (pores) are not random on the level of the individual cell. There are less of the extreme cells (with 6 long edges, or with 6 short edges) and more of the bland cells (with 3 short and 3 long edges) in the Finney packing than predicted on the basis of simple random expectations. Second, the distribution of cell classes within the packing is shown to be essentially homogeneously random. Evidence for extremely slight cell class clustering is found. The drainage and imbibition processes within the packing are simulated using pore-level algorithms. The algorithms utilise both the Haines' insphere approximation and the MS-P approximation for critical drainage meniscus curvature, and the cell cavity insphere radius approximation for critical imbibition meniscus curvature. Good agreement with experimental data is obtained, and the results confirm that drainage-imbibition hysteresis is a direct consequence of the connectivity between cells (pores), and is not an intrinsic property of the individual pore. Finally, the drainage and imbibition algorithms are adapted to emulate percolation theory models. The results prove that the classical bond problem of percolation theory does not adequately describe the drainage process for RCP, and that the classical site problem does not adequately describe the imbibition process for RCP.

Random field and other systems dominated by disorder fluctuations

Abstract: Spin-models in random fields (RFs) are good representations of many impure materials. Their macroscopic collective behaviour is dominated by the fluctuations in the random fields which accumulate on large scales even if the local field is arbitrarily small. This feature is shared by other weakly disordered models, like flux lines or domain walls in random media. We review some of the main theoretical attempts to describe such systems. A modification of Harris’ argument demonstrates that at the critical point the RF disorder is relevant and that (hyper)scaling must be changed. A domain argument invented by Imry and Ma shows that long-range order is not destroyed by weak RFs in more than d=2 dimensions. This result is supported both by a more refined treatment of the domain argument and by considering the roughness of an isolated domain wall due to the randomness. The wall (or flux line) becomes rough due to disorder but if d>2 the wall remains a well-defined object in RF systems. Different approaches are used to calculate the roughness exponent ζ for walls and lines. Some applications of ζ for the description of type-II superconductors and incommensurate systems are given. More detailed calculations are possible for one-dimensional, Bethe-lattice or the hierarchical Dyson model systems, which confirm as a rule the more approximate treatment of the other sections. In one dimension there is an interesting relation between the statistical mechanics of these models and nonlinear dynamics. Non-classical critical behaviour occurs in RF systems for d 0 is related to the violation of conventional hyperscaling and is determined by the energy ~H0ξ0 of a correlated region of size ξ. In a renormalization group treatment, the temperature T turns out to be a (dangerous) irrelevant variable which is the most prominent property of the systems considered in this review. The irrelevance of thermal fluctuations on large scales produces metastability and hysteresis effects both in the transition region and in the ordered phase, only briefly considered here. These features occur also in other systems with a disordered T=0 fixed point like in the ordered phase of a spin-glass.

Generalized Delta theorems for multivalued mappings

Abstract: The classical delta theorem can be generalized in a mathematically satisfying way to a broad class of multivalued and/or nonsmooth mappings, by examining the convergence in distribution of the sequence of difference quotients from the perspectives of recent developments in convergence theory for random closed sets and new descriptions of first-order behaviour of multivalued mappings. Such a theory opens the way to applications of asymptotic techniques in many areas of mathematical optimization where randomness and uncertainty play a role. Of special importance is the asymptotic convergence of measurable selections of multifunctions when the limit multifunction is single-valued almost surely.

Extension of an entropy property for binary input memoryless symmetric channels

Abstract: The channel output entropy property introduced by A.D. Wyner and J. Ziv (ibid., vol.IT-19, p.769-762, Nov.1973) for a binary symmetric channel is extended to arbitrary memoryless symmetric channels with binary inputs and discrete or continuous outputs. This yields lower bounds on the achievable information rates of these channels under constrained binary inputs. Using the interpretation of entropy as a measure of order and randomness, the authors deduce that output sequences of memoryless symmetric channels induced by binary inputs are of a higher degree of randomness if the redundancy of the input binary sequence is spread in memory rather than in one-dimensional asymmetry. It is of interest to characterize the general class of schemes for which this interpretation holds. >

Equivalence of generic equation method and the phenomenological model for linear transport problems in a two-state random scattering medium

Abstract: Linear particle transport problems in a two‐state random Markov medium are considered. Explicit equations are constructed that determine the expected particle density in a scattering medium. It is shown that the generic equation method is equivalent to the phenomenological model if the free flights of the particle are uncorrelated.

Fluctuations in Derrida's random energy and generalized random energy models

Abstract: The fluctuations of the finite-size corrections to the free energy per site of the random energy model (REM) and the generalized random energy model (GREM) are investigated. Almost sure behavior for the corrections of order (logN)/N is given. We also prove convergence in distribution for the corrections of order 1/N.

Theoretical Models for Polarimetric Microwave Remote Sensing of Earth Terrain

Abstract: Using the two-layer anisotropic random medium, a mathematically rigorous, fully polarimetric model is developed to compute the Mueller and covariance matrices in the backscattering direction for various kinds of earth terrain. The electric field is first written in the form of an integral equation involving the unperturbed dyadic Green's function in the absence of the permittivity fluctuations. The integral equation is then solved by an iterative series known as the Born series. With only the first term of the series, which physically describes a single scattering process, the fully polarimetric backscattering coefficients are derived. Four different kinds of upgoing and downgoing waves exist due to the excitation of both ordinary and extraordinary waves in the anisotropic random medium. An averaging scheme over the azimuthal direction is used to simulate the effects on the radar backscattering due to the azimuthal randomness in the growth direction of leaves in tree and grass fields.

The complexity of the stock market

Abstract: The stock market is a complex system, somewhere between the domains of order and randomness. Ordered systems are simple and predictable, and random systems are inherently unpredictable. Simple theories do not adequately describe security pricing, nor is pricing random. Rather, the market is permeated by a web of interrelated return effects. Substantial computational power is needed to disentangle, model, and exploit these return regularities.