Self-attracting walk on lattices

Abstract

We have studied a model of self-attracting walk proposed by Sapozhnikov using Monte Carlo method. The mean square displacement R 2 (t) ∼ t 2ν and the mean number of visited sites S(t) ∼ t k are calculated for one-, two-and three-dimensional lattice. In one dimension, the walk shows diffusive behaviour with ν = k = 1/2. However, in two and three dimension, we observed a non-universal behaviour, i.e., the exponent ν varies continuously with the strength of the attracting interaction. 1There is great interest in random walks and interacting walks [1,2,3]. The mean square displacement of a random walk, R 2 (t) follows a power-law R 2 (t) ∼ t 2ν . The ordinary random walk(RW) is diffusive, with ν = 1/2, in all dimensions. For a walk with repulsion, such as self-avoiding walk(SAW), the exponent ν is greater than 1/2. Random walk on a fractal is anomalous with ν < 1/2 [2,3]. Various models of interacting walks have been studied, such as true self-avoiding walk [4,5], generalized true SAW[6], the Domb-Joyce model [7], and an interacting walk with a weight factor p for each new site that the random walker visits [8,9]. A comparative study of interacting random walk models was performed by Duxbury et.al. [10,11].Recently, Sapozhikov proposed a generalized walk in which the probability for the walker to jump to a given site is proportional to p = exp(−nu), where n = 1 for the sites visited by the walker at least once and n = 0 for other sites [12]. If u < 0, the walker is attracted to its own trajectory. This walk is called a self-attracting walk(SATW). Monte Carlo studies have suggested that ν < 1/2 for u = −1 and u = −2 on two dimension and 1/4 < ν < 1/3 on three dimension [12]. However, Aarão Reis obtained non-universal behaviour of the SATW in one to four dimension using exact enumeration method [13]. Prasad et.al. concluded that the SATW in one dimension is diffusive [14].In the present comment we report results of a Monte Carlo simulation for the self-attracting walk in one, two and three dimension. We find that SATW in one dimension is diffusive. However, SATW in two and three dimension shows non-universal behaviour. Monte Carlo simulations were performed on one dimensional lattice (10 6 -steps with 2000-configurational averages for each parameter u), square lattice (10 6 -steps with 2000-averages), and cubic lattice (10 5 -steps with 2000-averages). The lattice sizes used in this simulation were L = 10 6 (1D), 1024×1024 (2D) and 200×200×200(3D). We always used the periodic boundary conditions. We calculated the mean square displacement R 2 (t) and the mean number of visited sites S(t) . Fig. 1 (a) shows the log-log plot of the mean square displacement against the time. We obtained the exponent by a least-square fit: ν = 0.500 (7) for u = 0, ν = 0.500(9) for u = −0.5, ν = 0.499(8) for u = −1.0, and ν = 0.498(9) for u = −2.0. All the lines are parallel in the large-time limit. These results means that SATW in one dimension is diffusive and support the conclusions of Prasad et.al.[14]. This diffusive behaviour is furth...

Fractal structure of a three-dimensional Brownian motion on an attractive plane

“…2. As can be seen from figure 2, for large values of α, since the problem reduces to the 2d random walk on the attractive plane, these two fractal dimensions converge to the same value close to the value ∼ 1.83 (this is comparable with the fractal dimension of the set of distinct sites visited by an 2d RW on a square lattice, deduced from the results reported in [12]). All error bars in this paper are estimated by using the standard least-squares analysis, and are almost of the same size as the symbols used in the figures.…”

Reinforced walks in two and three dimensions

“…( 1) but with u > 0 is unknown. On the other hand, there are several numerical studies of the TSATW with one-step reinforcement [6,7,8,9,10,11] p j = e κj u N j ′ =1 e κ j ′ u u > 0 .…”

Structural properties of self-attracting walks

“…1. The exponents k and ν of SATW have been found to depend on u [9,[17][18][19], although it was not clear for some time if ν and k decrease continuously with increasing u [18,19], or if a critical value u c exists, below, at, and above which the exponents show different universal behavior [9,17]. Recently, it was found by exhaustive computer simulations up to t = 5 • 10 9 time steps that there exists a swelling-collapse transition for SATW at a critical attraction u c [15], analogous to the Θ transition in linear polymers at temperature T = Θ when an attraction term exp(−A/T ), A < 0, is added to the self-avoiding constraint [12,13].…”