Directed paths in random media, function approximation and nonlinear regression

Directed paths in random media are used for function approximation and nonlinear regression. Input data are taken as random pinning. Wandering directed paths, subject to internal elasticity and interaction with the pinning, when pinned, are solutions to these problems.     

Directed paths on percolation clusters

We consider a variant of the problem of directed polymers on a disordered lattice, in which the disorder is geometrical in nature. In particular, we allow a finite probability for each bond to be absent from the lattice. We show, through the use of numerical and scaling arguments on both Euclidean and hierarchical lattices, that the model has two distinct scaling behaviors, depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold. We are particularly interested in the exponents and, defined by ft and xt , describing the free-energy and transverse fluctuations, respectively. Above the percolation threshold, the scaling behavior is governed by the standard random energy exponents (=1/3 and =2/3 in 1+1 dimensions). At the percolation threshold, we predict (and verify numerically in 1+1 dimensions) the exponents=1/2 and =v/v, where v and v are the directed percolation exponents. In addition, we predict the absence of a free phase in any dimension at the percolation threshold.     

Directed random walks in random environments

We use holding time methods to study the asymptotic behavior of pure birth processes with random transition rates. Both the normal and slow approaches to infinity are studied. Fluctuations are shown to obey the central limit theorem for almost all sample-transition rates. Our results are stronger, and our proofs simpler, then those of recently published studies.     

Directed self-organized critical patterns emerging from fractional Brownian paths

We discuss a family of clusters corresponding to the region whose boundary is formed by a fractional Brownian path y(i) and by the moving average function . Our model generates fractal directed patterns showing spatio-temporal complexity, and we demonstrate that the cluster area, length and duration exhibit the characteristic scaling behavior of SOC clusters. The function C_n(i) acts as a magnifying lens, zooming in (or out) the ‘avalanches’ formed by the cluster construction rule, where the magnifying power of the zoom is set by the value of the amplitude window n. On the basis of the construction rule of the clusters and of the relationship among the exponents, we hypothesize that our model might be considered to be a generalized stochastic directed model, including the Dhar–Ramaswamy (DR) model and the stochastic models as particular cases. As in the DR model, the growth and annihilation of our clusters are obtained from the set of intersections of two random walk paths, and we argue that our model is a variant of the directed self-organized criticality scheme of the DR model.     

Directed random polymers via nested contour integrals

We study the partition function of two versions of the continuum directed polymer in 1+1$1+1$ dimension. In the full-space version, the polymer starts at the origin and is free to move transversally in R$\mathbb{R}$, and in the half-space version, the polymer starts at the origin but is reflected at the origin and stays in R₋${\mathbb{R}}_{-}$. The partition functions solve the stochastic heat equation in full-space or half-space with mixed boundary condition at the origin; or equivalently the free energy satisfies the Kardar–Parisi–Zhang equation.     

Statistical mechanics of random paths on disordered lattices

The dependence of the universality class on the statistical weight of unrestricted random paths is explicitly shown both for deterministic and statistical fractals such as the incipient infinite percolation cluster. Equally weighted paths (ideal chain) and kinetically generated paths (random walks) belong, in general, to different universality classes. For deterministic fractals exact renormalization group techniques are used. Asymptotic behaviors for the end-to-end distance ranging from power to logarithmic (localization) laws are observed for the ideal chain. In all these cases, random walks in the presence of nonperfect traps are shown to be in the same universality class of the ideal chain. Logarithmic behavior is reflected insingular renormalization group recursions. For the disordered case, numerical transfer matrix techniques are exploited on percolation clusters in two and three dimensions. The two-point correlation function scales with critical exponents not obeying standard scaling relations. The distribution of the number of chains and the number of chains returning to the starting point are found to be well approximated by a log-normal distribution. The logmoment of the number of chains is found to have an essential type of singularity consistent with the log-normal distribution. A non-self-averaging behavior is argued to occur on the basis of the results.     

Average and most-probable photon paths in random media

Time-resolved experiments have revealed that, in contrast to the predictions of conventional diffusion theory, photons select certain curvilinear paths to travel between a source and a detector. Concepts of the average photon paths and Fermat paths are introduced on the basis of the non-Euclidean diffusion equation (NED) to explain experimental results. Comparison of the theory and the experiment demonstrates the potential of the NED to describe nondiffusive features of photon migration in the multiple-scattering regime.     

Glassy phases in a layered ising model with random interlayer exchange

Thermodynamics of a layered Ising model with infinite-range ferromagnetic intralayer interaction and random nearest-neighbor interlayer coupling is considered. A detailed analysis of the model with vanishing average interlayer coupling is presented. The Gibbs free energy is found in the critical region, and the existence of many metastable states is demonstrated. Thermodynamic parameters of the system are found for periodic states. As the mean square interlayer coupling increases, the equilibrium state of the system undergoes an infinite sequence of first-order phase transitions, the number of magnetic planes and the distance between them change discontinuously, and so do both bulk magnetization and magnetic susceptibility.     

Color image encoding in dual fractional Fourier-wavelet domain with random phases

A new cryptology in dual fractional Fourier-wavelet domain is proposed in this paper, which is calculated by discrete fractional Fourier transform and wavelet decomposition. Different random phases are used in different wavelet subbands in encryption. A new color image encoding method is also presented with basic color decomposition and encryption respectively. All the keys, including random phases and fractional orders in R, G and B three channels, should be correctly used in decryption, otherwise people cannot obtain the totally correct information. Some numerical simulations are presented to demonstrate the possibility of the method. It would have widely potential applications in digital color image processing and protection.     

Detecting unknown paths on complex networks through random walks

In this article, we investigate the problem of detecting unknown paths on complex networks through random walks. To detect a given path on a network a random walker should pass through the path from its initial node to its terminal node in turn. We calculate probability ?(t) that a random walker detects a given path on a connected network in t steps when it starts out from source node s. We propose an iteration formula for calculating ?(t). Generating function of ?(t) is also derived. Major factors affecting ?(t), such as walking time t, path length l, starting point of the walker, structure of the path, and topological structure of the underlying network are further discussed. Among these factors, two most outstanding ones are walking time t and path length l. On the one hand, ?(t) increases as t increases, and ?(∞)=1, which shows that the longer the walking time, the higher the chance of detecting a given path, and the walker will discover the path sooner or later so long as it keeps wandering on the network. On the other hand, ?(t) drops substantially as path length l increases, which shows that the longer the path, the lower the chance for the walker to find it in a given time. Apart from path length, path structure also has obvious effect on ?(t). Starting point of the walker has only minor influence on ?(t), but topological structure of the underlying network has strong influence on ?(t). Simulations confirm our analytic results.     

Sign phase transition and directed paths in random media

We consider a lattice model which corresponds to the high temperature expansions of disordered Ising and Heisenberg models and to the deeply localized regime of the disordered Anderson model. The spin correlation functions for the Ising and Heisenberg model and the amplitude of electron tunneling for the Anderson model exhibit a “sign phase transition.” At small concentration x of scatterers with a negative scattering amplitude these quantities have predictable signs while at large x their signs are unpredictable. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 4, 283–288 (25 August 1996) Published in English in the original Russian Journal. Edited by J. R. Anderson.     

Random paths and random surfaces on a digital computer

We describe a local stochastic process that generates statistical ensembles of paths (or surfaces) with a Boltzmann probabilistic weight. The method is expected to be applicable to Monte Carlo simulations of field theories that contain fermions, to the roughening transition and to a reformulation of gauge theories in terms of random surfaces.     

Full-band extended states with random phases in one-dimensional disordered system with specific impurities

We investigate transport properties of electrons in a one-dimensional (1D) disordered system consisting of a host chain attached with specific impurities. Every impurity, labelled by j and possessing site energy , is side-coupled to two adjacent sites of the host chain with hopping integral t _1j and changesthe original nearest-neighbor (NN) hopping to t _2j . We show that if and for all impurities, with t ₀ being the NN hopping of the host chain, the states in the whole band are extended, even though s and positions of impurities are random. The phases of these states, however, are spatially random, corresponding to finite free path and infinite localization length in such a 1D system.Received: 6 May 2004, Published online: 12 July 2004PACS: 72.15.Rn Localization effects (Anderson or weak localization) - 72.80.Ng Disordered solids - 73.20.Jc Delocalization processes     

Universal properties of shortest paths in isotropically correlated random potentials

We consider the optimal paths in a d-dimensional lattice, where the bonds have isotropically correlated random weights. These paths can be interpreted as the ground state configuration of a simplified polymer model in a random potential. We study how the universal scaling exponents, the roughness and the energy fluctuation exponent, depend on the strength of the disorder correlations. Our numerical results using Dijkstra's algorithm to determine the optimal path in directed as well as undirected lattices indicate that the correlations become relevant if they decay with distance slower than 1/r in d = 2 and 3. We show that the exponent relation 2ν - ω = 1 holds at least in d = 2 even in case of correlations. Both in two and three dimensions, overhangs turn out to be irrelevant even in the presence of strong disorder correlations. Received 20 December 2002 / Received in final form 10 April 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: schorr@lusi.uni-sb.de     

Directed polymers in a random environment: Some results on fluctuations

We consider a polymer model on ℤ ₊ ^d where to each edgee is associated a random variable v(e). A polymer configuration is represented by a directed pathr and has a weight exp[-β ∑ _e ∈r ν(e)], withβ=1/T the inverse temperature. We extend some rigorous results that have been obtained for the ground state of this model to finite temperatures. In particular we obtain some upper and lower bounds on sample-to-sample free energy fluctuations, and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations     

Probability distribution of random paths in the Ising model at low temperature

Communications in Mathematical Physics - The random walk representation of then-dimensional Ising model exhibits the 2-point correlation function 〈σ(x) σ(y)〉 as a sum of...