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 $l\ge K$. In this Letter 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. We find evidence that the existence of a collapse transition depends sensitively on the details of the model and has an unexpected dependence on dimension.

• Received 17 March 2006

DOI:https://doi.org/10.1103/PhysRevLett.96.240603