Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d ≥ 3

Abstract

The problems considered in the present paper have their roots in two different cultures. The `true’ (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit et al. (Phys Rev B 27:1635–1645, 1983). This is a nearest neighbor non-Markovian random walk in \({{\mathbb Z}^d}\) which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (Probab Theory Relat Fields 92:337–349, 1992) (independently of the physics community), is the continuous space–time counterpart: a diffusion in \({{\mathbb R}^d}\) pushed by the negative gradient of the (mollified) occupation time measure of the process. In both cases, similar long memory effects are caused by a path-wise self-repellency of the trajectories due to a push by the negative gradient of (softened) local time. We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrent dimensions. First, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle. The main results are diffusive limits. In the case of TSAW, for a wide class of self-interaction functions, we establish diffusive lower and upper bounds for the displacement and for a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. In the case of SRBP, we prove full CLT without restrictions on the interaction functions. These results settle part of the conjectures, based on non-rigorous renormalization group arguments (equally ‘valid’ for the TSAW and SRBP cases), in Amit et al. (1983). The proof of the CLT follows the non-reversible version of Kipnis–Varadhan theory. On the way to the proof, we slightly weaken the so-called graded sector condition.