Random walks avoiding their convex hull with a finite memory

Abstract

Fix integers $d\ge 2$ and $k\ge d-1$. Consider a random walk $X_0,X_1,\dots$ in ${\mathbb{R}}^d$ in which, given $X_0,X_1,\dots ,X_n$ ($n\ge k$), 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 $\{0,X_{n-k},\dots ,X_n\}$ only at $X_n$. For $k=\infty$ 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 $8/(9{\pi }^2)$.

31 pages, 3 figures; v2: minor revisions

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• Superdiffusive planar random walks with polynomial space-time drifts