A Metric Graph for Which the Number of Possible Endpoints of a Random Walk Grows Minimally

TL;DR

This paper identifies the specific structure of metric graphs that minimizes the growth in the number of possible endpoints for a random walk, revealing that such graphs are unions of linear paths emanating from a single vertex.

Contribution

The paper proves that the minimal growth of possible random walk endpoints occurs in metric graphs formed by multiple linear paths sharing a common vertex.

Findings

Minimal endpoint growth occurs in graphs with a star-like structure

Union of linear paths from a single vertex minimizes endpoint expansion

Provides a characterization of graphs with minimal random walk endpoint growth

Abstract

We prove that metric graph with the minimal growth of the number of possible endpoints of a random walk is the union of several linear paths coming out of the same vertex