Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty

We consider a weakly self-avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp[-β|t-s|^-p] where t and s are the times at which the common site is visited and p is a parameter. We prove that if p<1 and β is sufficiently large, then the walk behaves ballistically, i.e., the distance to the end of the walk grows linearly with the number of steps in the walk. We also give a heuristic argument that if p>3/2, then the walk should have diffusive behavior. The proof and the heuristic argument make use of a real-space renormalization group transformation.