Random Walk with Barycentric Self-interaction

Abstract

We study the asymptotic behaviour of a d-dimensional self-interacting random walk (X n ) n∈N (N := {1, 2, 3, . . .}) which is repelled or attracted by the centre of mass G n = n −1 n i=1 X i of its previous trajectory. The walk's trajectory (X 1 , . . . , X n ) models a random polymer chain in either poor or good solvent. In addition to some natural regularity conditions, we assume that the walk has one-step mean driftfor ρ ∈ R and β ≥ 0. When β < 1 and ρ > 0, we show that X n is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: n −1/(1+β) X n converges almost surely to some random vector. When β ∈ (0, 1) there is sub-ballistic rate of escape. For β ≥ 0, ρ ∈ R we give almost-sure bounds on the norms X n , which in the context of the polymer model reveal extended and collapsed phases.Analysis of the random walk, and in particular of X n − G n , leads to the study of real-valued time-inhomogeneous non-Markov processes (Z n ) n∈N on [0, ∞) with mean drifts of the formwhere β ≥ 0 and ρ ∈ R. The study of such processes is a time-dependent variation on a classical problem of Lamperti; moreover, they arise naturally in the context of the distance of simple random walk on Z d from its centre of mass, for which we also give an apparently new result. We give a recurrence classification and asymptotic theory for processes Z n satisfying (0.1), which enables us to deduce the complete recurrence classification (for any β ≥ 0) of X n − G n for our self-interacting walk.

Excursions and path functionals for stochastic processes with asymptotically zero drifts

“…Upper bounds similar to those in Corollary 3.2 can be derived from [36, Section 3]: see Theorem 2.4 of [8] for a similar application of such results, albeit under more restrictive assumptions. As far as the authors are aware, the lower bounds in Corollary 3.2 are new.…”

“…The upper bounds in Section 4 of [36] essentially apply in the present setting (concretely, use [36,Theorem 3.2] with Lemma 4.1 here), and lead to slightly sharper upper bounds than those in our Theorem 2.6. (See also Section 6 of [8] for some variations on these upper bounds.) However, the lower bounds in [36] cannot readily be applied here, even assuming a uniform bound on the increments of X t .…”

Random walks avoiding their convex hull with a finite memory

“…A broad class of models is provided by random walks (or diffusions) that interact with the occupation measure of their past trajectory. This interaction can be local, such for reinforced [21] or excited random walks [5], in which the walker's motion is biased by its occupation measure in the immediate vicinity, or global, such as for processes with self-interaction mediated via some global functional of the past trajectory, such as a centre of mass or other occupation statistic [4,9,18,20,[26][27][28]. In either case, the selfinteraction can be attractive, corresponding to the collapsed polymer phase, or repulsive, corresponding to the extended phase.…”

“…using (9). Moreover, the integrand in the final integral in (10) is positive, by definition of A + .…”

Co-evolving agents subject to local versus nonlocal barycentric interactions

“…Exactly solvable MF models have been proposed in different contexts including portfolio theory [12], coupled phase oscillators [7], traffic dynamics [5,16], self-propelled organisms [26], etc. Related MF models with barycentric self-interactions are studied [2] and [10]. Moreover, the class of models discussed in the present paper is amenable to a discrete velocity Boltzmann equation of the Ruijgrok-Wu type [25] and can be hence analytically discussed.…”