THE DIFFUSIVE PHASE OF A MODEL OF SELF-INTERACTING WALKS

We consider simple random walk on Z(d) perturbed by a factor exp[beta T-(p)J(T)], where T is the length of the walk and J(T) = Sigma(0 less than or equal to i<j less than or equal to T)delta(w(i),w(j)). For p = 1 and dimensions d greater than or equal to 2, we prove that this walk behaves diffusively for all -infinity < beta < beta(0), with beta(0) > 0 For d > 2 the diffusion constant is equal to 1, but for d = 2 it is renormalized, For d = 1 and p = 3/2, we prove diffusion for all real beta (positive or negative). For d > 2 the scaling limit is Brownian motion, but for d less than or equal to 2 it is the Edwards model (with the ''wrong'' sign of the coupling when beta > 0) which governs the limiting behaviour; the latter arises since for p = 4-d/2, T-(p)J(T) is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.