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Durham University

Department of Mathematical Sciences

Academic Staff

Publication details for Mikhail Menshikov

Comets, Francis, Menshikov, Mikhail V., Volkov, Stanislav & Wade, Andrew R. (2011). Random walk with barycentric self-interaction. Journal of Statistical Physics 143(5): 855-888.

Author(s) from Durham


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 Gn=n−1∑ni=1Xi 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 drift E[Xn+1−Xn∣Xn−Gn=x]≈ρ∥x∥−βx^ for 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. When β≥0 and 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 form E[Zn+1−Zn∣Zn=x]≈ρx−β−xn, (0.1) where 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.