The diffusive phase of a model of self-interacting walks

Abstract

We consider simple random walk onZ d perturbed by a factor exp[βT −P J T], whereT is the length of the walk and $$J_T = \sum\nolimits_{0 \leqslant i< j \leqslant T} \delta _{\omega (i),\omega (j)} $$ . Forp=1 and dimensionsd≥2, we prove that this walk behaves diffusively for all − ∞ 0. Ford>2 the diffusion constant is equal to 1, but ford=2 it is renormalized. Ford=1 andp=3/2, we prove diffusion for all real β (positive or negative). Ford>2 the scaling limit is Brownian motion, but ford≤2 it is the Edwards model (with the “wrong” sign of the coupling when β>0) which governs the limiting behaviour; the latter arises since for $$p = \frac{{4 - d}}{2}$$ ,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.