True self-attracting walk

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 I or 0 in the original SATW model. And n(i,j) takes n(s) for t(v) greater than or equal to 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, proportional to t(2v) and proportional to 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. (C) 2001 Elsevier Science B.V. All rights reserved.