Leila Sloman writes about five mathematicians at ETH Zurich who, while pursuing a different problem in late 2025, stumbled upon a simple proof of the supercritical sharpness conjecture for all infinite transitive graphs—a decades-old question in percolation theory about how quickly networks flood past a critical threshold. The proof confirms that above the critical probability, fluid covers nearly the entire graph, resolving what researchers had called "the one remaining fortress" in the field.
- Percolation theory originated from Rosalind Franklin's 1940s work on coal porosity at the British Coal Utilization Research Association
- Oded Schramm, who co-initiated the study of percolation on transitive graphs with Itai Benjamini, died in a hiking fall in 2008 at age 46
- The key insight was reordering a standard "sprinkling" technique—analyzing the sprinkled edges first rather than last—which both simplified the proof and made it general enough for all transitive graphs
- The subcritical half of the sharpness conjecture had already been proved in 2007 by Antunović and Veselić
- A major open question remains: what happens exactly at the critical probability on three-dimensional lattices
- Percolation theory originated from Rosalind Franklin's 1940s work on coal porosity at the British Coal Utilization Research Association
- Oded Schramm, who co-initiated the study of percolation on transitive graphs with Itai Benjamini, died in a hiking fall in 2008 at age 46
- The key insight was reordering a standard "sprinkling" technique—analyzing the sprinkled edges first rather than last—which both simplified the proof and made it general enough for all transitive graphs
- The subcritical half of the sharpness conjecture had already been proved in 2007 by Antunović and Veselić
- A major open question remains: what happens exactly at the critical probability on three-dimensional lattices
Seong-Gyun Im, Taewoo Kang, and S. Joon Kwon introduce a new method called Hilbert entropy to quantify the complexity of high-dimensional data by using space-filling curves like the Hilbert curve. This approach aims to reduce dimensions while preserving context, offering an alternative to traditional metrics such as Lyapunov exponents or fractal dimension that often fail to capture the intrinsic nature of complex physical systems. The authors validated this methodology through its ability to accurately identify critical phenomena and phase transitions in percolation models and spin models.
- Validated via high concordance with theoretical phase transition points.
- Demonstrates a potential linear relationship between scaling exponents and Euclidean dimensions for scale-invariant geometries.
- Applicable to 2D and 3D geometrical analysis of complex systems.
- Validated via high concordance with theoretical phase transition points.
- Demonstrates a potential linear relationship between scaling exponents and Euclidean dimensions for scale-invariant geometries.
- Applicable to 2D and 3D geometrical analysis of complex systems.
