Continuum Percolation

Continuum percolation is the study of connectivity in random geometric graphs where nodes are placed in continuous space and edges exist when nodes fall within a fixed distance of each other. Unlike lattice percolation, which lives on a rigid grid, continuum percolation models systems where spatial proximity is the only organizing principle — wireless sensor networks, disease transmission in mobile populations, or the overlap of tree canopies in a forest. The canonical model, the Gilbert disk model, places points according to a Poisson process and connects any two points within distance r; the phase transition at a critical density is the mathematical foundation for understanding coverage and connectivity in infrastructure-free networks. The striking result is that geometric constraints can produce percolation thresholds that differ dramatically from their lattice counterparts, meaning that real-world spatial networks cannot be approximated by grid models without systematic error.