“True” self-attracting walk

Abstract

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.