Topological data analysis of continuum percolation with disks

Area of Science:

• Mathematics
• Topology
• Statistical Physics

Background:

• Continuum percolation is a fundamental concept in statistical physics, describing the formation of connected clusters in random geometric systems.
• Understanding percolation transitions is crucial for various fields, including materials science, network theory, and fluid dynamics.

Purpose of the Study:

• To investigate continuum percolation with disks using advanced topological data analysis tools.
• To characterize the topological features of disk configurations and their evolution.
• To identify topological invariants associated with the percolation transition.

Main Methods:

• Interpreting each realization of disk percolation as a topological subspace of a 2D Euclidean space.
• Applying persistent homology, a technique from topological data analysis, to quantify topological features.
• Systematically varying the number and radius of disks to observe topological changes.

Main Results:

• Persistent homology successfully revealed topological changes in disk configurations.
• Evidence suggests that the longest-persisting topological invariant appears at or near the percolation transition.
• The study establishes a link between topological features and the critical phenomena of percolation.

Conclusions:

• Topological data analysis provides novel insights into continuum percolation phenomena.
• Persistent homology can effectively detect and characterize topological invariants related to phase transitions.
• The findings contribute to a deeper understanding of geometric and topological properties in random systems.