On Sharp Thresholds in Random Geometric Graphs

TL;DR

This paper characterizes sharp threshold phenomena for vertex-monotone properties in Poisson random geometric graphs and applies this to demonstrate a sharp threshold for satisfiability in a geometric random k-SAT model.

Contribution

It provides a new characterization of sharp thresholds in geometric graphs and applies it to establish a sharp threshold in geometric random k-SAT.

Findings

Characterization of vertex-monotone properties with sharp thresholds.

Demonstration of a sharp satisfiability threshold in geometric random k-SAT.

Application of geometric graph theory to satisfiability problems.

Abstract

We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.