Self-attracting Walk Research Articles

Trapping in self-avoiding walks with nearest-neighbor attraction.

The statistics of self-avoiding random walks have been used to model polymer physics for decades. A self-avoiding walk that grows one step at a time on a lattice will eventually trap itself, which occurs after an average of 71 steps on a square lattice. Here, we consider the effect of nearest-neighbor attractive interactions on isolated growing self-avoiding walks, and we examine the effect that self-attraction has both on the statistics of trapping as well as on chain statistics through the transition between expanded and collapsed walks at the theta point. We find that the trapping length increases exponentially with the nearest-neighbor contact energy, but that there is a local minimum in trapping length for weakly self-attractive walks. While it has been controversial whether growing self-avoiding walks have the same asymptotic behavior as traditional self-avoiding walks, we find that the theta point is not at the same location for growing self-avoiding walks, and that the persistence length converges much more rapidly to a smaller value.

Measuring site fidelity and spatial segregation within animal societies.

Summary Animals often display a marked tendency to return to previously visited locations that contain important resources, such as water, food, or developing brood that must be provisioned. A considerable body of work has demonstrated that this tendency is strongly expressed in ants, which exhibit fidelity to particular sites both inside and outside the nest. However, thus far many studies of this phenomena have taken the approach of reducing an animal's trajectory to a summary statistic, such as the area it covers.Using both simulations of biased random walks, and empirical trajectories from individual rock ants, Temnothorax albipennis, we demonstrate that this reductive approach suffers from an unacceptably high rate of false negatives.To overcome this, we describe a site‐centric approach which, in combination with a spatially‐explicit null model, allows the identification of the important sites towards which individuals exhibit statistically significant biases.Using the ant trajectories, we illustrate how the site‐centric approach can be combined with social network analysis tools to detect groups of individuals whose members display similar space‐use patterns.We also address the mechanistic origin of individual site fidelity; by examining the sequence of visits to each site, we detect a statistical signature associated with a self‐attracting walk – a non‐Markovian movement model that has been suggested as a possible mechanism for generating individual site fidelity.

Self-attracting walk on heterogeneous networks.

Understanding human mobility in cyberspace becomes increasingly important in this information era. While human mobility, memory-dependent and subdiffusive, is well understood in Euclidean space, it remains elusive in random heterogeneous networks like the World Wide Web. Here we study the diffusion characteristics of self-attracting walks, in which a walker is more likely to move to the locations visited previously than to unvisited ones, on scale-free networks. Under strong attraction, the number of distinct visited nodes grows linearly in time with larger coefficients in more heterogeneous networks. More interestingly, crossovers to sublinear growths occur in strongly heterogeneous networks. To understand these phenomena, we investigate the characteristic volumes and topology of the cluster of visited nodes and find that the reinforced attraction to hubs results in expediting exploration first but delaying later, as characterized by the scaling exponents that we derive. Our findings and analysis method can be useful for understanding various diffusion processes mediated by human.

Non-random walks in monkeys and humans

Principles of self-organization play an increasingly central role in models of human activity. Notably, individual human displacements exhibit strongly recurrent patterns that are characterized by scaling laws and can be mechanistically modelled as self-attracting walks. Recurrence is not, however, unique to human displacements. Here we report that the mobility patterns of wild capuchin monkeys are not random walks, and they exhibit recurrence properties similar to those of cell phone users, suggesting spatial cognition mechanisms shared with humans. We also show that the highly uneven visitation patterns within monkey home ranges are not entirely self-generated but are forced by spatio-temporal habitat heterogeneities. If models of human mobility are to become useful tools for predictive purposes, they will need to consider the interaction between memory and environmental heterogeneities.

Scaling of human behavior during portal browsing

We investigated flows of visitors migrating between different portal subpages. Two various portals were studied as weighted networks where nodes are portal subpages and edge weights are numbers of user transitions. Such networks differ from networks of portal subpages connected by hyperlinks prepared by portal designers. Distributions of link weights, node strengths, and times spent by visitors at one subpage follow power laws over several decades for data collected during two different days and for weekly data. The distribution of numbers P(z) of unique subpages visited during one session is exponential and there is a square-root dependence between the total number of transitions n during a single visit and the average z . A model of portal surfing is developed where the browsing process corresponds to a self-attracting walk on the weighted network with a short memory. Results of numerical simulation are in agreement with weekly and daily portal data, and our analytical approach fits empirical data in the center part of scaling regime.

Grand-canonical and canonical solution of self-avoiding walks with up to three monomers per site on the Bethe lattice

We solve a model of polymers represented by self-avoiding walks on a lattice, which may visit the same site up to three times in the grand-canonical formalism on the Bethe lattice. This may be a model for the collapse transition of polymers where only interactions between monomers at the same site are considered. The phase diagram of the model is very rich, displaying coexistence and critical surfaces, critical, critical end point, and tricritical lines, as well as a multicritical point. From the grand-canonical results, we present an argument to obtain the properties of the model in the canonical ensemble, and compare our results with simulations in the literature. We do actually find extended and collapsed phases, but the transition between them, composed by a line of critical end points and a line of tricritical points, separated by the multicritical point, is always continuous. This result is at variance with the simulations for the model, which suggest that part of the line should be a discontinuous transition. Finally, we discuss the connection of the present model with the standard model for the collapse of polymers (self-avoiding, self-attracting walks), where the transition between the extended and collapsed phases is a tricritical point.

RECURRENCE OF SELF-REPELLING AND SELF-ATTRACTING WALKS ON THE PRE-SIERPIŃSKI GASKET AND ℤ

We prove recurrence of self-repelling and self-attracting walks on the pre-Sierpiński gasket and ℤ. We also determine a set of parameters for which this walk defines infinite, σ-finite and ergodic measure preserving dynamical systems.

SELF-ATTRACTING WALK ON NON-UNIFORM SUBSTRATES

Self-attracting walk (SATW) on non-uniform substrates has been investigated by Monte Carlo simulations. The non-uniform substrates are described by Leath percolation clusters with occupied probability p. p stands for the degree of non-uniformity, and takes on values in the range pc≲p ≤1 where pc is the threshold of percolation. For the case of strong attractive interaction u, p has little influence on the walk which is dominated by attractive interactions. Furthermore, in the case of small scales, the exponent ν of the mean end-to-end distance <R2(t)> versus time t is given by ν≃1/(ds+1), while the exponent k of visited sites versus t is given by k≃ds/(ds+1), where ds are the fractal dimensions of the substrates. For u ≃ 0, the walk reduces to the random walk on percolations with p in pc≲p≤1. Also, ν and k decrease sensitively with the reduction of p. It is found, the blocked sites in the substrates (i.e. defects) have much greater influence on the walk driven by thermal flunctuation than that dominated by the attractive interaction.

“True” self-attracting walk

Abstract The self-attracting walk (SATW) model was extended and investigated by Monte Carlo simulations. In the extended two-dimensional model, the environment parameter n i , j takes the value of visited times t v at the site ( i , j ) when t v is less than a saturated environment parameter n s , instead of the alternative of 1 or 0 in the original SATW model. And n i , j takes n s for t v ⩾ n s . There exists a critical attractive interaction u c depending on n s and long time scaling relations of mean square displacement and mean visited sites, 〈 R 2 〉∝ t 2 v and 〈 S 〉∝ t k . For u > u c , v =1/3 and k =2/3. But for u u c , v =1/2 and k =0.9.

Structural properties of self-attracting walks.

Self-attracting walks (SATW) with attractive interaction u>0 display a swelling-collapse transition at a critical u(c) for dimensions d>or=2, analogous to the Theta transition of polymers. We are interested in the structure of the clusters generated by SATW below u(c) (swollen walk), above u(c) (collapsed walk), and at u(c), which can be characterized by the fractal dimensions of the clusters d(f) and their interface d(I). Using scaling arguments and Monte Carlo simulations, we find that for u<u(c), the structures are in the universality class of clusters generated by simple random walks. For u>u(c), the clusters are compact, i.e., d(f)=d and d(I)=d-1. At u(c), the SATW is in a new universality class. The clusters are compact in both d=2 and d=3, but their interface is fractal: d(I)=1.50+/-0.01 and 2.73+/-0.03 in d=2 and d=3, respectively. In d=1, where the walk is collapsed for all u and no swelling-collapse transition exists, we derive analytical expressions for the average number of visited sites <S> and the mean time <t> to visit S sites.

Swelling-collapse transition of self-attracting walks

We study the structural properties of self-attracting walks in d dimensions using scaling arguments and Monte Carlo simulations. We find evidence of a transition analogous to the Theta transition of polymers. Above a critical attractive interaction u(c), the walk collapses and the exponents nu and k, characterizing the scaling with time t of the mean square end-to-end distance <R2> approximately t(2nu) and the average number of visited sites <S> approximately t(k), are universal and given by nu=1/(d+1) and k=d/(d+1). Below u(c), the walk swells and the exponents are as with no interaction, i.e., nu=1/2 for all d, k=1/2 for d=1 and k=1 for d>/=2. At u(c), the exponents are found to be in a different universality class.

Reply to the comment by J W Lee. Self-attracting walk: are the exponents universal?

Problems concerning the value of the exponent in the relationship describing a self-attracting walk are considered. It is suggested that further study is needed to understand how the exponent depends on the coupling energy of the walk.

Self-attracting walk on lattices

We have studied a model of self-attracting walk proposed by Sapozhnikov using the Monte Carlo method. The mean-square displacement and the mean number of visited sites are calculated for one-, two- and three-dimensional lattices. In one dimension, the walk shows diffusive behaviour with . However, in two and three dimensions, we observed a non-universal behaviour, i.e. the exponent varies continuously with the strength of the attracting interaction.

Diffusive behaviour of self-attractive walks

We study an attractive random walk called the SATW in one dimension and show that the walk is diffusive. We show rigorously that asymptotically , where is the average of the square of the number of distinct points visited by the walk in N steps. Since , the average square of the distance to the origin, is of the same order as we get the result that asymptototically.

Non-universal behaviour of self-attracting walks

A recently proposed model for self-attracting walks is studied using exact enumeration techniques. The probability of a step is proportional to exp(-nu), where n=1 for sites already visited by the walker and n=0 for the others, with u<0. Series for the mean-square displacement (RN2) approximately N2 nu and the mean number of visited sites (SN) approximately Ns of N-step walks are calculated in one to four dimensions. In all dimensions anomalous diffusion is observed, and exponents nu and s vary continuously with the strength parameter u: The results are compared with simulations and with previous results for static and dynamic models of generalized random walks in one dimension. The behaviour in two and three dimensions may describe anomalous diffusion in real systems.

Self-attracting walk with nu &lt;1/2

A model of a self-attracting walk is proposed. Mean square displacement of a particle xi grows with time t as a power law xi 2 approximately t2 nu with the exponent nu <1/2. It is shown that nu =1/(2D-Db) where D is the fractal dimension of the cluster consisting of the visited sites and Db is the fractal dimension of its boundary. A compact cluster with fractal boundaries was obtained by a computer simulation for d=2. The derived value of nu is in a good agreement with that obtained by computer simulation.

Statistics of the true self-avoiding walk in one dimension

Numerical results of a Monte Carlo study for the true self-avoiding walk in one dimension are presented. For any positive value of the strength parameter (0<g< infinity ), the root-mean-square displacement RN and the range SN of the walk are characterised by two universal exponents nu =2/3+or-0.003 and s=2/3+or-0.01 respectively. For negative g (self-attracting walk) both RN and SN exhibit a saturation effect: limN to infinity RN=Rinfinity (g) and limN to infinity SN=Sinfinity (g) with Rinfinity (g) approximately (-g)-1 and Sinfinity (g) approximately (-g)-1 at g<or approximately=0. A simple scaling analysis in N and g is proposed and found to be consistent with the Monte Carlo results.