Forward Jump Random Walk on a Cycle Graph and Its Hitting Time

This paper presents an investigation into a random walk on a cycle graph with restricted forward movement at most $m$ steps, known as the forward jump random walk. The study derives precise formulas for the probability mass function of the arriving state, the hitting time, and its expected value and variance. These formulas are obtained using a combinatorial method as an alternative to the eigenvector-based approach commonly used. The findings reveal that the distributions, expected value, and variance of the hitting time are influenced by the maximum jump length, the number of vertices or states, and the number of walking steps. Notably, as the maximum jump length or number of walking steps increases, the probability mass function of the arriving state converges to a uniform distribution, indicating equal visitation of all states in the long run. Furthermore, larger jumps result in increased average hitting time, while the size of the graph has minimal effect on it