Asymptotics of the number of possible endpoints of a random walk on a directed Hamiltonian metric graph

TL;DR

This paper derives the leading asymptotic behavior of the number of potential endpoints for a random walk on a directed Hamiltonian metric graph, relevant for understanding wave packet propagation on such structures.

Contribution

It provides the first asymptotic formula for endpoint counts of random walks on directed Hamiltonian metric graphs, linking graph structure to dynamical behavior.

Findings

Derived the leading asymptotic term for endpoint counts

Connected graph dynamics to wave packet propagation

Established foundational results for directed Hamiltonian metric graphs

Abstract

In this paper, the leading term of the asymptotics of the number of possible final positions of a random walk on a directed Hamiltonian metric graph is found. Consideration of such dynamical systems could be motivated by problems of propagation of narrow wave packets on metric graphs.