The Range of a Random Walk on a Comb

The graph obtained from the integer grid Z x Z by the removal of all horizontal edges that do not belong to the x-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex v, in the next step it will visit one of the neighbors of v, each with probability 1/d(v), where d(v) denotes the degree of v. We answer a question of Csaki, Csorgo, Foldes, Revesz, and Tusnady by showing that the expected number of vertices visited by a random walk on the comb after n steps is (1/2 root 2 pi + o(1)) root n log n. This contradicts a claim of Weiss and Havlin

Some results and problems for anisotropic random walks on the plane

This is an expository paper on the asymptotic results concerning path behaviour of the anisotropic random walk on the two-dimensional square lattice Z^2. In recent years Mikl\'os and the authors of the present paper investigated the properties of this random walk concerning strong approximations, local times and range. We give a survey of these results together with some further problems.Comment: 20 page

Two Phase Transitions for the Contact Process on Small Worlds

In our version of Watts and Strogatz's small world model, space is a d-dimensional torus in which each individual has in addition exactly one long-range neighbor chosen at random from the grid. This modification is natural if one thinks of a town where an individual's interactions at school, at work, or in social situations introduces long-range connections. However, this change dramatically alters the behavior of the contact process, producing two phase transitions. We establish this by relating the small world to an infinite "big world" graph where the contact process behavior is similar to the contact process on a tree.Comment: 24 pages, 6 figures. We have rewritten the phase transition in terms of two parameters and have made improvements to our original result

Slow Encounters of Particle Pairs in Branched Structures

On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous structures this property does not hold and, although a single random walker will certainly return to its starting point, two moving particles may never meet. This striking property has been shown to hold, for instance, on infinite combs. Due to the huge variety of natural phenomena which can be modeled in terms of encounters between two (or more) particles diffusing in comb-like structures, it is fundamental to investigate if and, if so, to what extent similar effects may take place in finite structures. By means of numerical simulations we evidence that, indeed, even on finite structures, the topological inhomogeneity can qualitatively affect the two-particle problem. In particular, the mean encounter time can be polynomially larger than the time expected from the related one particle problem.Comment: 8 pages, 12 figures; accepted for publication in Physical Review

Random walks on combs

We develop techniques to obtain rigorous bounds on the behaviour of random walks on combs. Using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length. We also calculate exactly the spectral dimension of some fixed non-translationally invariant combs. We relate the spectral dimension to the critical exponent of the mass of the two-point function for random walks on random combs, and compute mean displacements as a function of walk duration. We prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension.Comment: 42 pages, 4 figure

Prediction of sustained harmonic walking in the free-living environment using raw accelerometry data

Objective. Using raw, sub-second level, accelerometry data, we propose and validate a method for identifying and characterizing walking in the free-living environment. We focus on the sustained harmonic walking (SHW), which we define as walking for at least 10 seconds with low variability of step frequency. Approach. We utilize the harmonic nature of SHW and quantify local periodicity of the tri-axial raw accelerometry data. We also estimate fundamental frequency of observed signals and link it to the instantaneous walking (step-to-step) frequency (IWF). Next, we report total time spent in SHW, number and durations of SHW bouts, time of the day when SHW occurred and IWF for 49 healthy, elderly individuals. Main results. Sensitivity of the proposed classification method was found to be 97%, while specificity ranged between 87% and 97% and prediction accuracy between 94% and 97%. We report total time in SHW between 140 and 10 minutes-per-day distributed between 340 and 50 bouts. We estimate the average IWF to be 1.7 steps-per-second. Significance. We propose a simple approach for detection of SHW and estimation of IWF, based on Fourier decomposition. The resulting approach is fast and allows processing of a week-long raw accelerometry data (approx. 150 million measurements) in relatively short time (~half an hour) on a common laptop computer (2.8 GHz Intel Core i7, 16 GB DDR3 RAM)