## Electronic Communications in Probability

### Random Walks that Avoid Their Past Convex Hull

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 8 (2003), paper no. 2, 6-16.

Dates
Accepted: 16 February 2003
First available in Project Euclid: 18 May 2016

https://projecteuclid.org/euclid.ecp/1463608886

Digital Object Identifier
doi:10.1214/ECP.v8-1065

Mathematical Reviews number (MathSciNet)
MR1961285

Zentralblatt MATH identifier
1009.60085

Rights

#### Citation

Angel, Omer; Benjamini, Itai; Virág, Bálint. Random Walks that Avoid Their Past Convex Hull. Electron. Commun. Probab. 8 (2003), paper no. 2, 6--16. doi:10.1214/ECP.v8-1065. https://projecteuclid.org/euclid.ecp/1463608886

#### References

• B. Davis, Reinforced random walk. Probab. Theory Related Fields 84 (1990), no. 2, 203–229.
• G. Lawler, Intersections of random walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1991.
• R. Pemantle, Random processes with reinforcement. (Preprint)
• O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221–288.
• 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.
• B. Tóth and W. Werner, The true self-repelling motion. Probab. Theory Related Fields 111 (1998), no. 3, 375–452.