On Random Walks In A Plane

Abstract

Consider a random walk in a plane in which a particle at any stage moves one unit in any one of the four directions, namely, north, south, east, west with equal probability. The problem of finding the distribution of any characteristic of the above random walk when the particle reaches a fixed point $(a,b)$ after $d$ steps reduces to the counting of lattice paths in a plane in which the path can move one unit in any of the four directions. In this paper, path counting results related to the boundaries $y-x=k_1$ and $y+x=k_2$ such as touchings, crossings, etc., are obtained by using either combinatorial or probabilistic methods. Some extensions to higher dimensions are indicated.