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 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 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 nonrigorous renormalization group arguments (equally 'valid' for the TSAW and SRBP 123 692 I. Horváth et al. 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. cases), in

Superdiffusive Bounds on Self-repellent Brownian Polymers and Diffusion in the Curl of the Gaussian Free Field in d=2

“…Computations very similar to those in [21] and [6] show that the Gaussian measure π defined by the expectations and covariances…”

“…It has been proved in Sect. 3.3 of [6] that the Gaussian probability measure π is stationary and ergodic for the Markov process η t . That is: if the initial vector field F is sampled from this distribution, then the vector field profile seen from the position of the moving particle will have the same distribution at any later time.…”

Knowledge acquisition: A Complex networks approach

“…The established link between these two concepts (random walks and diffusion), paved the way to the development of the theoretical approaches to understand the characteristics of matter, such as the concept of Brownian motion [23], leading to the discovering of several macroscopic characteristics and the modeling of many real-world phenomena.…”

Superdiffusive bounds on self-repellent precesses in d=2 — extended abstract

“…For results in d = 1 see [9], [13], [12], [8] and the survey [10]. For results in d ≥ 3 see [3]. The present lecture concentrates on recent results on the d = 2 case.…”

“…It was proved in [8] (for d = 1) and [3] (for arbitrary d) that the Gaussian measure π on Ω defined by the covariances…”