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

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 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.

*© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2017*