Random walks avoiding their convex hull with a finite memory

Abstract

Fix integers d ≥ 2 and k ≥ d − 1. Consider a random walk X 0 , X 1 , . . . in R d in which, given X 0 , X 1 , . . . , X n (n ≥ k), the next step X n+1 is uniformly distributed on the unit ball centred at X n , but conditioned that the line segment from X n to X n+1 intersects the convex hull of {0, X n−k , . . . , X n } only at X n . For k = ∞ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-k model, and comment on some open problems. In the case where d = 2 and k = 1, we obtain the limiting speed explicitly: it is 8/(9π 2 ). δ) .If k = 1 this ends the proof. Otherwise, onas before. Hence, on G n ∩ {B ⊆ Π(X n )},≥ δ 2d Vol 2d (B 1 × B 2 ) (Vol d B(0; δ)) 2 .Iterating this argument gives the result.

Variations of the elephant random walk

“…which, together with Proposition A.1(i), tells us thatE(T n X n+1 ) = (2p − 1)/3 1 − (2p − 1)/3 + o(1) = 2p − 1 2(2 − p) + o(1) as n → ∞. (A 5). …”

“…Let us mention Chen and Renshaw [3], for example, who investigated a walk in dimension d and the probability of returns. Menshikov and Volkov [17] considered continuous-time processes generalizing the ERW and questions of transience and recurrence, and Comets et al [5] studied a kind of self-avoiding walk in R d . See also the literature cited therein for further references.…”