ABSTRACT
The process starts at zero and transitions either one step to the right with probability p or one step to the left with probability 1-p. If a random walk occurs on a finite subset of integers, it is termed a finite-state random walk or a random walk on a finite grid or a bordered random walk. In this chapter, the authors also consider some variations of a random walk, such as two-, three-, and higher-dimensional random walks, and random walks on graphs. Some variations of a simple random walk include a random walk with loops (or delays), in the sense that the allowed moves are to the right with probability p, to the left with probability q, and remaining in the same state with probability 1-p-q. A two-dimensional random walk is recurrent if and only if it is symmetric. Hence, a bug crawling randomly along a railroad track will visit every inch of it infinitely many times.
