### On maximizing the speed of a random walk in fixed environments

**Amichai Lampert**

*(NA)*

**Assaf Shapira**

*(Technion – Israel Institute of Technology)*

#### Abstract

We consider a random walk in a fixed $\mathbb{Z}$ environment composed of two point types: $q$-drifts (in which the probabiliy to move to the right is $q$, and $1-q$ to the left) and $p$-drifts, where $\frac{1}{2}<q<p$. We study the expected hitting time of a random walk at $N$ given the number of $p$-drifts in the interval $[1,N-1]$, and find that this time is minimized asymptotically by equally spaced $p$-drifts.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-9

Publication Date: May 30, 2013

DOI: 10.1214/ECP.v18-2431

#### References

- Procaccia, Eviatar B.; Rosenthal, Ron. The need for speed: maximizing the speed of random walk in fixed
environments.
*Electron. J. Probab.*17 (2012), no. 13, 19 pp. MR2892326 - Zeitouni, Ofer. Random walks in random environment.
*Lectures on probability theory and statistics,*189--312, Lecture Notes in Math., 1837,*Springer, Berlin,*2004. MR2071631

This work is licensed under a Creative Commons Attribution 3.0 License.