Random walk on a disordered directed Bethe lattice

The random walk of a particle on a directed Bethe lattice of constant coordinanceZ is examined in the case of random hopping rates. As a result, the higher the coordinance, the narrower the regions of anomalous drift and diffusion. The annealed and quenched mean square dispersions are calculated in all dynamical phases. In opposition to the one-dimensional (Z=2) case, the annealed and quenched mean quadratic dispersions are shown to be identical in all phases.We shall employ indifferently the expressions Bethe lattice or infinite Cayley tree to denote an infinite ramified lattice of constant coordinanceZ.^{(4, 5)}     

On a resonance energy model based on expansion in terms of acyclic moments: Exact results

Summary The new concept of the resonance energy in conjugated hydrocarbons introduced by Jiang Y, Zhang H (1989) Theor Chim Acta 75:279 is further developed. This model is based on expansion of the -electron energy in terms of moments which are also equal to numbers of closed walks in a molecular graph. The reference system is established by counting only acyclic walks, i.e. those tracing only on acyclic subgraphs. Because acyclic walks could be counted only up to some finite length, the energy of the reference system has been evaluated by truncating higher terms in the expansion. In this paper a finite expression for the energy of the same reference system is derived, thus allowing its exact evaluation. The exact values differ significantly from the truncated ones. This difference, as well as the discrepancy between exact results and chemical experience, are discussed.     

One-dimensional Ising chain with competing interactions: Exact results and connection with other statistical models

We study the ground state properties of a one-dimensional Ising chain with a nearest-neighbor ferromagnetic interactionJ ₁, and akth neighboranti-ferromagnetic interactionJ _k . WhenJ _k/J₁=–1/k, there exists a highly degenerate ground state with a residual entropy per spin. For the finite chain with free boundary conditions, we calculate the degeneracy of this state exactly, and find that it is proportional to the (N+k–1)th term in a generalized Fibonacci sequence defined by,F _N ^(k) =F _N–1 ^(k) +F _N–k ^(k) . In addition, we show that this one-dimensional model is closely related to the following problems: (a) a fully frustrated two-dimensional Ising system with a periodic arrangement of nearest-neighbor ferro- and antiferromagnetic bonds, (b) close-packing of dimers on a ladder, a 2× strip of the square lattice, and (c) directed self-avoiding walks on finite lattice strips.Work partially supported by grants from AFOSR and ARO.     

Photon channelling in foams

Experiments by Gittings, Bandyopadhyay and Durian (Europhys. Lett. 65, 414 (2004)) demonstrate that light possesses a higher probability to propagate in the liquid phase of a foam due to total reflection. The authors term this observation photon channelling which we investigate in this article theoretically. We first derive a central relation in the work of Gitting et al. without any free parameters. It links the photon's path-length fraction f in the liquid phase to the liquid fraction ɛ. We then construct two-dimensional Voronoi foams, replace the cell edges by channels to represent the liquid films and simulate photon paths according to the laws of ray optics using transmission and reflection coefficients from Fresnel's formulas. In an exact honeycomb foam, the photons show superdiffusive behavior. It becomes diffusive as soon as disorder is introduced into the foams. The dependence of the diffusion constant on channel width and refractive index is explained by a one-dimensional random-walk model. It contains a photon channelling state that is crucial for the understanding of the numerical results. At the end, we shortly comment on the observation that photon channelling only occurs in a finite range of ɛ.     

Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance

Let ξ₁,ξ₂,... be independent random variables with distributions F₁F₂,... in a triangular array scheme (F _i may depend on some parameter). Assume that Eξ _i = 0, Eξ _i ² < ∞, and put \(S_n = \sum {_{i = 1}^n \;} \xi _i ,\;\overline S _n = \max _{k \leqslant n} S_k\). Assuming further that some regularly varying functions majorize or minorize the “averaged” distribution \(F = \frac{1}{n}\sum {_{i = 1}^n F_i }\), we find upper and lower bounds for the probabilities P(S _n > x) and \(P(\bar S_n > x)\). We also study the asymptotics of these probabilities and of the probabilities that a trajectory {S _k } crosses the remote boundary {g(k)}; that is, the asymptotics of P(max_k≤n(S _k ? g(k)) > 0). The case n = ∞ is not excluded. We also estimate the distribution of the first crossing time.     

Different self-avoiding walks on percolation clusters: A small-cell real-space renormalization-group study

We calculate the average number of stepsN for edge-to-edge, normal, and indefinitely growing self-avoiding walks (SAWs) on two-dimensional critical percolation clusters, using the real-space renormalization-group approach, with small H cells. Our results are of the formN=AL ^D _SAW+B, whereL is the end-to-end distance. Similarly to several deterministic fractals, the fractal dimensionsD _SAW for these three different kinds of SAWs are found to be equal, and the differences between them appear in the amplitudesA and in the correction termsB. This behavior is atributed to the hierarchical nature of the critical percolation cluster.     

The Renormalization Group and Optimization of Entropy

We illustrate the possible connection that exists between the extremal properties of entropy expressions and the renormalization group (RG) approach when applied to systems with scaling symmetry. We consider three examples: (1) Gaussian fixed-point criticality in a fluid or in the capillary-wave model of an interface; (2) Lévy-like random walks with self-similar cluster formation; and (3) long-ranged bond percolation. In all cases we find a decreasing entropy function that becomes minimum under an appropriate constraint at the fixed point. We use an equivalence between random-walk distributions and order-parameter pair correlations in a simple fluid or magnet to study how the dimensional anomaly at criticality relates to walks with long-tailed distributions.     

Dynamic exponent in extremal models of pinning

The depinning transition of a front moving in a time-independent random potential is studied. The temporal development of the overall roughness w(L,t) of an initially flat front, , is the classical means to have access to the dynamic exponent. However, in the case of front propagation in quenched disorder via extremal dynamics, we show that the initial increase in front roughness implies an extra dependence over the system size which comes from the fact that the activity is essentially localized in a narrow region of space. We propose an analytic expression for the exponent and confirm this for different models (crack front propagation, Edwards-Wilkinson model in a quenched noise etc.). Received 27 August 1999     

Random Walks on Trees and an Inequality of Means

We define trees generated by bi-infinite sequences, calculate their walk-invariant distribution and the speed of a biased random walk. We compare a simple random walk on a tree generated by a bi-infinite sequence with a simple random walk on an augmented Galton-Watson tree. We find that comparable simple random walks require the augmented Galton-Watson tree to be larger than the corresponding tree generated by a bi-infinite sequence. This is due to an inequality for random variables with values in [1, [ involving harmonic, geometric and arithmetic mean.     

Numerical Simulations of Random Walk in Random Environment

This paper is concerned with the numerical simulation of a random walk in a random environment in dimension d = 2. Consider a nearest neighbor random walk on the 2-dimensional integer lattice. The transition probabilities at each site are assumed to be themselves random variables, but fixed for all time. This is the random environment. Consider a parallel strip of radius R centered on an axis through the origin. Let X _R be the probability that the walk that started at the origin exits the strip through one of the boundary lines. Then X _R is a random variable, depending on the environment. In dimension d = 1, the variable X _R converges in distribution to the Bernoulli variable, X = 0, 1 with equal probability, as R . Here the 2-dimensional problem is studied using Gauss-Seidel and multigrid algorithms.