Speed of Convergence and Moderate Deviations of FPP on Random Geometric Graphs

TL;DR

This paper analyzes the speed of convergence and moderate deviations in first-passage percolation on random geometric graphs, focusing on properties like geodesic paths, fluctuations, and spanning trees in the supercritical regime.

Contribution

It provides a quantitative shape theorem and characterizes fluctuations and properties of spanning trees in first-passage percolation on random geometric graphs.

Findings

Established a shape theorem for FPP on random geometric graphs.

Characterized fluctuations and moderate deviations of geodesic paths.

Analyzed properties of spanning trees and semi-infinite paths.

Abstract

This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations, and fluctuations, aiming to establish a quantitative shape theorem. Furthermore, we examine fluctuations in geodesic paths and characterize the properties of spanning trees and their semi-infinite paths.