Random Walks that Avoid Their Past Convex Hull

Omer Angel (Weizmann Institute of Science)
Itai Benjamini (Weizmann Institute of Science)
Bálint Virág (MIT)

Abstract


We explore planar random walk conditioned to avoid its past convex hull. We prove that it escapes at a positive lim sup speed. Experimental results show that fluctuations from a limiting direction are on the order of $n^{3/4}$. This behavior is also observed for the extremal investor, a natural financial model related to the planar walk.

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Pages: 6-16

Publication Date: February 16, 2003

DOI: 10.1214/ECP.v8-1065

References

  1. B. Davis, Reinforced random walk. Probab. Theory Related Fields 84 (1990), no. 2, 203--229. Math. Review 91a:60179
  2. G. Lawler, Intersections of random walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1991. Math. Review 92f:60122
  3. R. Pemantle, Random processes with reinforcement. (Preprint) No Math Review number available.
  4. O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221--288. Math. Review 2001m:60227
  5. B. Tóth, Self-interacting random motions -- a survey. In: Random walks -- A Collection of Surveys. Eds: P. Révész and B. Tóth. Bolyai Society Mathematical Studies, 9, Budapest, 1998. Math. Review 2001d:60048
  6. B. Tóth and W. Werner, The true self-repelling motion. Probab. Theory Related Fields 111 (1998), no. 3, 375--452. Math. Review 99i:60092
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