On a Type of Self-Avoiding Random Walk with Multiple Site Weightings and   Restrictions
  We introduce a new class of models for polymer collapse, given by random walks on regular lattices which are weighted according to multiple site visits. A Boltzmann weight $\omega_l$ is assigned to each $(l+1)$-fold visited lattice site, and self-avoidance is incorporated by restricting to a maximal number $K$ of visits to any site via setting $\omega_l=0$ for $l\geq K$. In this paper we study this model on the square and simple cubic lattices for the case K=3. Moreover, we consider a variant of this model, in which we forbid immediate self-reversal of the random walk. We perform simulations for random walks up to $n=1024$ steps using FlatPERM, a flat histogram stochastic growth algorithm. Unexpectedly, we find evidence that the existence of a collapse transition depends sensitively on the details of the model. 
J Krawczyk, T Prellberg, AL Owczarek, A Rechnitzer