Symmetric Walks on Paths and Cycles

Abstract

SummaryWe study a symmetric simple random walk on a finite section of the integer lattice (with various end conditions) and on the vertices of a polygon whose nodes are labeled as . We study the probability distributions (specifically the expectation and the variance) of the time until first return to the starting position, the number of nodes visited during this time, the time until all nodes are visited, the last node to be visited, the time to return to the starting position after visiting all nodes and the number of nodes visited in the interim. These questions are answered using elementary methods readily understood by college mathematics students—methods such as symmetry, recursive relations and mathematical induction.