Random walk with barycentric self-interaction
Résumé
We study the asymptotic behaviour of a d-dimensional self-interacting random walk (X-n)(n subset of N) (N := {1, 2, 3, ... }) which is repelled or attracted by the centre of mass G(n) = n(-1) Sigma(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 drift
E[Xn+1 - X-n vertical bar X-n - G(n) = x] approximate to rho parallel to x parallel to(-beta)(x) over cap
for rho epsilon R and beta >= 0. When beta < 1 and rho > 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+beta)) X-n converges almost surely to some random vector. When beta epsilon (0, 1) there is sub-ballistic rate of escape. When beta >= 0 and rho epsilon R we give almost-sure bounds on the norms parallel to X-n parallel to, 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 epsilon N) on [0,infinity) with mean drifts of the form (0.1)