Random patterns generated by random permutations of natural numbers

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time n, whose moves to the right or to the left are prescribed by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site X at time n, obtain the probability measure of different excursions and define the asymptotic distribution of the number of “U-turns" of the trajectories - permutation “peaks" and “through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers 1,2,3, ...,L on sites of a 1d or 2d lattices containing L sites. We calculate the distribution function of the number of local “peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss surprising collective behavior emerging in this model.     

Slowly moving matter-wave gap soliton propagation in weak random optical lattices

We systematically investigate slowly moving matter-wave gap soliton propagation in weak random optical lattices. With the weak randomness, an effective-particle theory is constructed to show that the motion of a gap soliton is similar to a particle moving in random potentials. Based on the effective-particle theory, the effects of the randomness on gap solitons are obtained and the trajectories of gap solitons are well predicted. Moreover, the general laws that describe the movement depending on the weak randomness are obtained. We find that with an increase of the random strength, the ensemble-average velocity reduces slowly and the reflection probability becomes larger. The theoretical results based on the effective-particle theory are confirmed by the numerical simulations based on the Gross-Pitaevskii equation.     

Estimation of Probability Densities of the Absolute Value and Argument of Complex Random Variables

We particularize criteria proposed by one of the authors for estimating the probability densities of a random vector of size N to the case of even N. For the presentation convenience in this case, two-dimensional vectors are treated as complex random variables and the probability densities of their absolute values and arguments are estimated. The estimation algorithms are synthesized on the basis of the obtained criteria under conditions of both presence and absence of complete a priori information on the sample statistic. A number of examples are given to prove that the obtained algorithms are optimal in the statistical sense. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 48, No. 7, pp. 616–627, July 2005.     

Slowly moving matter--wave gap soliton propagation in weak random nonlinear potential

We systematically investigate the motion of slowly moving matter--wave gap solitons in a nonlinear potential, produced by the weak random spatial variation of the atomic scattering length. With the weak randomness, we construct an effective-particle theory to study the motion of gap solitons. Based on the effective-particle theory, the effect of the randomness on gap solitons is obtained, and the motion of gap solitons is finally solved. Moreover, the analytic results for the general behaviours of gap soliton motion, such as the ensemble-average speed and the reflection probability depending on the weak randomness are obtained. We find that with the increase of the random strength the ensemble-average speed of gap solitons decreases slowly where the reduction is proportional to the variance of the weak randomness, and the reflection probability becomes larger. The theoretical results are in good agreement with the numerical simulations based on the Gross--Pitaevskii equation.     

Application of Wavelet Transformation in Analysis of Multiplicity Phase Space

Using the wavelet transformation, the probability distribution for the multiplicity phase space which is obtained by simulation from random cascading α model is discussed. We find that using the wavelet transformation the probability moments and the correlation moments for the experimental data of the multiplicity phase space probability distribution canbe directly analysed. An actual calculation mathod is suggested.     

Self-segregation in chemical reactions,diffusion in a catalytic environment and an ideal polymer near a defect

We study a family of equivalent continuum models in one dimension. All these models map onto a single equation and include simple chemical reactions, diffusion in presence of a trap or a source and an ideal polymer chain near an attractive or repulsive site. We have obtained analytical results for the survival probability, total growth rate, statistical properties of nearest-neighbour distribution between a trap and unreacted particle and mean-squared displacement of the polymer chain. Our results are compared with the known asymptotic results in the theory of discrete random walks on a lattice in presence of a defect.     

Random versus realistic interactions for low-lying nuclear spectra

We compare the shell-model results for realistic interactions with those obtained for various ensembles of random matrix elements. We show that, although the quantum numbers of the ground states in the even-even nuclei have a high probability ( approximately 60%) to be J(pi)T = 0(+)0, the overlap of those states with the realistic wave functions is very small in average. The transition probabilities B(E2) predicted with random interactions are also too small. The presence of the regular pairing is shown to be a significant element of realistic physics not reproduced by random interactions.     

Taming Griffiths' singularities: Infinite differentiability of quenched correlation functions

We prove infinite differentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic field in the presence of Griffiths' singularities. We also show uniqueness of the Gibbs state and exponential decay of truncated correlation functions with probability one. Our results are obtained through new simple modified high temperature or low activity expansions whose convergence can be displayed by elementary probabilistic arguments. Our results require no assumptions on the probability distributions of the random parameters, except for the obvious one of no percolation of infinite couplings, and, in the strong field situation, for the also obvious requirement that zero magnetic fields do not percolate.Partially supported by the CNPq and FAPESP.Partially supported by the NSF under grants DMS-9208029 and INT-9016926.Partially supported by the CNPq.     

Integro-differential equation for joint probability density in phase space associated with continuous-time random walk

We derive an integro-differential equation for the joint probability density function in phase space associated with the continuous-time random walk, with generic waiting time probability density function and external force. This equation permits us to investigate whole diffusion processes covering initial-, intermediate-, and long-time ranges, which can distinguish the evolution details for systems having the same behavior in the long-time limit with different initial- and intermediate-time behaviors. Moreover, we obtained analytic solutions for probability density functions both in velocity and phase spaces, and interesting dynamic behaviors are discovered.     

Information dynamics: temporal behavior of uncertainty measures

We carry out a systematic study of uncertainty measures that are generic to dynamical processes of varied origins, provided they induce suitable continuous probability distributions. The major technical tools are the information theory methods and inequalities satisfied by Fisher and Shannon information measures. We focus on the compatibility of these inequalities with the prescribed (deterministic, random or quantum) temporal behavior of pertinent probability densities.      

Persistence in random walk in composite media

We consider a class of inhomogeneous media known as composite media that is often encountered in experimental sciences, and investigate the persistence probability of a random walker in such a system. Analytical and numerical results for the crossover time scales are obtained for a composite system with two homogeneous components and three homogeneous components respectively.     

A generalization of random walk models to correlations over two jumps

Using published results on continuous time random walk theories, we show that the random walk theory of Gissler and Rother is equivalent to a master equation with jumps to further neighbor sites. We extend the theory to include time correlations over two jumps. No special assumptions are made in the analysis, so that the theory may be applied to any lattice type with a general time probability distribution for jumps; a generalized second-order differential equation is given for the results. In the special case of an exponential time probability density, a simple homogeneous second order differential equation is obtained which is shown to be equivalent to a certain two-state master equation model.     

Random walks on lattices with randomly distributed traps I. The average number of steps until trapping

For a random walk on a lattice with a random distribution of traps we derive an asymptotic expansion valid for smallq for the average number of steps until trapping, whereq is the probability that a lattice point is a trap. We study the case of perfect traps (where the walk comes to an end) and the extension obtained by letting the traps be imperfect (i.e., by giving the walker a finite probability to remain free when stepping on a trap). Several classes of random walks of varying dimensionality are considered and special care is taken to show that the expansion derived is exact up to and including the last term calculated. The numerical accuracy of the expansion is discussed.     

Classical broadcasting is possible with arbitrarily high fidelity and resolution

We quantify the resolution with which any probability distribution may be distinguished from a displaced copy of itself in terms of a characteristic width. This width, which we call the resolution, is well defined for any normalizable probability distribution. We use this concept to study the broadcasting of classical probability distributions. Ideal classical broadcasting creates two (or more) output random variables each of which has the same distribution as the input random variable. We show that the universal broadcasting of probability distributions may be achieved with arbitrarily high fidelities for any finite resolution. By restricting probability distributions to any finite resolution we have therefore shown that the classical limit of quantum broadcasting is consistent with the actual classical case.     

Statistical properties of N random sinusoidal waves in additive Gaussian noise

The first order probability density functions of the sums of N independent sinusoidal waves having random amplitudes and phases in additive Gaussian noise are studied for the cases where N is fixed and where N in Poisson distributed. The conditional moments about the origin are obtained in closed form or both situations. The corresponding probability density functions for the envelope are also studied. The even conditional moments about the origin are also obtained in closed form. Representative numerical results are presented.     

Potentials for almost Markovian random fields

A positive almost Markovian random field is a probability measure on a lattice gas whose finite set conditional probabilities are continuous and positive. We show that each such random field has a potential and in the translation invariant case an absolutely convergent potential. We give a criterion for determining which random fields correspond to pair potentials, or in generaln-body potentials. We show that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.     

Law of addition in random matrix theory

We discuss the problem of adding random matrices, which enables us to study Hamiltonians consisting of a deterministic term plus a random term. Using a diagrammatic approach and introducing the concept of “gluon connectedness”, we calculate the density of energy levels for a wide class of probability distributions governing the random term, thus generalizing a result obtained recently by Brézin, Hikami and Zee. The method used here may be applied to a broad class of problems involving random matrices.     

On stochastic complex beam-beam interaction models with Gaussian colored noise

This paper is to continue our study on complex beam-beam interaction models in particle accelerators with random excitations Y. Xu, W. Xu, G.M. Mahmoud, On a complex beam-beam interaction model with random forcing [Physica A 336 (2004) 347-360]. The random noise is taken as the form of exponentially correlated Gaussian colored noise, and the transition probability density function is obtained in terms of a perturbation expansion of the parameter. Then the method of stochastic averaging based on perturbation technique is used to derive a Fokker-Planck equation for the transition probability density function. The solvability condition and the general transforms using the method of characteristics are proposed to obtain the approximate expressions of probability density function to order ε.Also the exact stationary probability density and the first and second moments of the amplitude are obtained, and one can find when the correlation time equals to zero, the result is identical to that derived from the Stratonovich-Khasminskii theorem for the same model under a broad-band excitation in our previous work.     

Detecting and filtering a flow of different pulsed signals against the noise background

We consider the problem of separate detection and filtration of a flow of pulsed signals. We assume that statistically independent pulsed signals of different shape and random amplitude appear randomly against the background of noise. An algorithm for estimation of the detection probability of each signal is obtained. Recurrent expressions for thea posteriori probability density of each signal are found. The results of numerical simulation of the proposed algorithms of signal processing are presented. Lobachevsky State University of Nizhny Novgorod Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 1, pp. 92–100, January, 1999.     

Return probability: Exponential versus Gaussian decay

We analyze, both analytically and numerically, the time-dependence of the return probability in closed systems of interacting particles. Main attention is paid to the interplay between two regimes, one of which is characterized by the Gaussian decay of the return probability, and another one is the well-known regime of the exponential decay. Our analytical estimates are confirmed by the numerical data obtained for two models with random interaction. In view of these results, we also briefly discuss the dynamical model which was recently proposed for the implementation of a quantum computation.