Eur. Phys. J. B (2017) 90: 174
https://doi.org/10.1140/epjb/e2017-80279-0

Regular Article

Alzheimer random walk^*

¹ Research Institute for Science Education, Inc., Kitaku, 603-8346 Kyoto, Japan
² Department of Physics, Tokyo Denki University, Hikigun, 350-0394 Saitama, Japan

^a e-mail: t.odagaki@kb4.so-net.ne.jp

Received: 17 May 2017
Published online: 18 September 2017

Abstract

Using the Monte Carlo simulation, we investigate a memory-impaired self-avoiding walk on a square lattice in which a random walker marks each of sites visited with a given probability p and makes a random walk avoiding the marked sites. Namely, p = 0 and p = 1 correspond to the simple random walk and the self-avoiding walk, respectively. When p> 0, there is a finite probability that the walker is trapped. We show that the trap time distribution can well be fitted by Stacy’s Weibull distribution $\mathit{b}{{}^{\mathrm{\right(}}\frac{\mathit{a}}{\mathit{b}}^{\mathrm{\left)}}}^{\frac{\mathit{a}\mathrm{+}\mathrm{1}}{\mathit{b}}}\mathrm{\left[}\mathit{\Gamma }\mathrm{\right(}\frac{\mathit{a}\mathrm{+}\mathrm{1}}{\mathit{b}}\mathrm{)}{\mathrm{]}}^{-1}{\mathit{x}}^{\mathit{a}}\exp {}^{\mathrm{(}}\mathrm{-}\frac{\mathit{a}}{\mathit{b}}{\mathit{x}}^{\mathit{b}}{}^{\mathrm{)}}$ where a and b are fitting parameters depending on p. We also find that the mean trap time diverges at p = 0 as ~p^{− α} with α = 1.89. In order to produce sufficient number of long walks, we exploit the pivot algorithm and obtain the mean square displacement and its Flory exponent ν(p) as functions of p. We find that the exponent determined for 1000 step walks interpolates both limits ν(0) for the simple random walk and ν(1) for the self-avoiding walk as [ ν(p) − ν(0) ] / [ ν(1) − ν(0) ] = p^β with β = 0.388 when p ≪ 0.1 and β = 0.0822 when p ≫ 0.1.