Trapping in Growing Self-Avoiding Walks: Numerical and Exact Results

Self-avoiding walks on lattices are used to model the statistics of polymer chains. Here we consider growing self-avoiding walks (GSAWs) on a lattice, which grow by taking their Nth step into a randomly chosen unoccupied site adjacent to the N-1th step. It is known from simulations that on a square lattice, a GSAW will become trapped after a mean of 71 steps, but this is a purely empirical fact. We have extended the square lattice GSAW to include nearest-neighbor attractive interactions, similar to those used to model polymers in poor solvents, which lead to a non-monotonic trend in the mean trapping length. To gain additional insight into the statistics of GSAW trapping, we consider simplified cases of geometrically restricted lattices that are two to three sites wide. Using recursion relations and generating functions, we are able to derive exact expressions for parameters such as the mean trapping length, the asymptotic behavior of the trapping probability distribution, and the effect of nearest-neighbor attraction, finding, for example, that walks on a restricted square lattice are trapped after a mean of exactly 17 steps. Our findings provide mathematical insight into a phenomenon that has been known only empirically since the 1980s.