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Random walks avoiding their convex hull with a finite memory | | Francis Comets
; Mikhail V. Menshikov
; Andrew R. Wade
; | | Date: |
26 Feb 2019 | | Abstract: | Fix integers $d geq 2$ and $kgeq d-1$. Consider a random walk $X_0, X_1,
ldots$ in $mathbb{R}^d$ in which, given $X_0, X_1, ldots, X_n$ ($n geq k$),
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 ${0, X_{n-k}, ldots, X_n}$ 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 $8/(9pi^2)$. | | Source: | arXiv, 1902.9812 | | Services: | Forum | Review | PDF | Favorites | |
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