Distribution of the k-Multiple Point Range in the Closed Simple Random Walk I

TL;DR

This paper analyzes the distribution of points visited multiple times in a closed simple random walk using a graph approach, deriving moments and the joint generating function for one dimension.

Contribution

It introduces a graph-based method to derive the distribution and moments of the k-multiple point range in closed simple random walks, providing explicit formulas for one dimension.

Findings

Derived a general expression for moments of the k-multiple point range.

Calculated and analyzed the joint generating function for one-dimensional walks.

Provided insights into the distribution behavior for large walk lengths.

Abstract

The distribution of the number of points of the closed simple random walk, visited a given number of times (the k-multiple point range) is analysed by a graph based approach. A general expression for the moments is derived. In this paper the joint generating function for dimension one is completely calculated and analysed for large lengths.