  The statistics of self-avoiding random walks have been used to model polymer
physics for decades. A self-avoiding walk that grows one step at a time on a
lattice will eventually trap itself, which occurs after an average of 71 steps
on a square lattice. Here, we consider the effect of nearest-neighbor
attractive interactions on the growing self-avoiding walk, and examine the
effect that self-attraction has both on the statistics of trapping as well as
on chain statistics through the transition between expanded and collapsed walks
at the theta point. We find the trapping length increases exponentially with
the nearest-neighbor contact energy, but that there is a local minimum in
trapping length for weakly self-attractive walks. While it has been
controversial whether growing self-avoiding walks have the same asymptotic
behavior as traditional self-avoiding walks, we find that the theta point is
not at the same location for growing self-avoiding walks, and that the
persistence length converges much more rapidly to a smaller value.
