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 ∕ ( 9 π 2 ) .