Shortest Path through Random Points

TL;DR

This paper proves that the normalized length of power-weighted shortest paths through randomly sampled points on a Riemannian manifold converges to a scaled Riemannian distance under a modified metric, revealing a deep connection between random paths and geometry.

Contribution

It establishes a rigorous convergence result linking shortest path lengths through random points to a modified Riemannian metric, extending understanding of geometric properties of random graphs.

Findings

Normalized shortest path length converges almost surely.

Limit is a constant multiple of the Riemannian distance.

Convergence involves a metric tensor modified by the sampling density.

Abstract

Let $(M,g_1)$ be a complete $d$-dimensional Riemannian manifold for $d > 1$. Let $\mathcal X_n$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. We prove that the normalized length of the power-weighted shortest path between $x, y$ through $\mathcal X_n$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_p = f^{2(1-p)/d} g_1$, where $p > 1$ is the power parameter.