On maximizing the speed of a random walk in fixed environments. (English) Zbl 1319.60092

Lampert, Amichai; Shapira, Assaf

Electron. Commun. Probab. 18, Paper No. 40, 9 p. (2013).

Summary: 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.

MSC:

60G50

Sums of independent random variables; random walks

Keywords:

random walk; speed maximization; fixed environments