Random walks avoiding their convex hull with a finite memory

Fix integers d≥2 and k≥d−1. Consider a random walk X0,X1,… in Rd in which, given X0,X1,…,Xn (n≥k), the next step Xn+1 is uniformly distributed on the unit ball centred at Xn, but conditioned that the line segment from Xn to Xn+1 intersects the convex hull of {0,Xn−k,…,Xn} only at Xn. For k=∞ this is...

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Bibliographic Details
Published in:Indagationes mathematicae 2020-01, Vol.31 (1), p.117-146
Main Authors: Comets, Francis, Menshikov, Mikhail V., Wade, Andrew R.
Format: Article
Language:English
Subjects:
Ballistic
Convex hull
Mathematics
Rancher
Random walk
Self-avoiding
Speed
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Summary:Fix integers d≥2 and k≥d−1. Consider a random walk X0,X1,… in Rd in which, given X0,X1,…,Xn (n≥k), the next step Xn+1 is uniformly distributed on the unit ball centred at Xn, but conditioned that the line segment from Xn to Xn+1 intersects the convex hull of {0,Xn−k,…,Xn} only at Xn. 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∕(9π2).
ISSN:0019-3577
1872-6100
DOI:10.1016/j.indag.2019.11.002