Mathematicians are building a precise, continuous model of the random, tangled geometry that emerges in two-dimensional physical systems like percolation—the process of fluid seeping through a porous material. This project addresses a fundamental gap in statistical mechanics: while physicists have long known that certain random structures (such as the clusters formed in a critical percolation model) become "conformally invariant" at large scales, the exact mathematical description of their shape and internal distances has remained out of reach. The work builds on recent breakthroughs—including the Schramm-Loewner evolution (SLE) and the Brownian map—to construct a canonical metric for these random surfaces. If successful, the research will solve de Gennes' "Ant in a Labyrinth" problem, a decades-old question about how a random walker moves through a disordered medium like a critical percolation network. This is fundamental science with no immediate practical application, but the mathematical tools developed could eventually inform algorithms for network routing, material design, or image analysis—much as earlier work on random geometry unexpectedly underpinned modern data compression and error-correcting codes.
View original technical description
In the last 25 years, there has been spectacular progress in the mathematical study of the random planar geometric structures which arise from statistical mechanics models in two dimensions. Important developments include: - Schramm's invention of the Schramm-Loewner evolution (SLE), the canonical model of a conformally invariant random curve. - Lawler-Schramm-Werner's theory of conformal restriction and its application to the study of planar Brownian motion. - Smirnov's proof of the conformal invariance of critical percolation and the Ising model. - Le Gall and Miermont's construction of the Brownian map, the canonical continuous model of a uniformly random surface. - Duplantier-Sheffield's rigorous proof of the KPZ formula for Liouville quantum gravity, the canonical model of a random two-dimensional Riemannian manifold. - Kupiainen-Rhodes-Vargas's construction and analysis of Liouville conformal field theory. The purpose of this project is to study the metric and analytical aspects of the random planar geometric structures which can be built from SLEs. The goal is to use these results to answer long-standing questions about the discrete models from statistical mechanics whose scaling limits are described by SLEs and related processes. The key objectives include: - Construct the canonical conformally covariant planar metric which describes the scaling limit of the intrinsic metric for critical planar statistical mechanics models, including percolation. - Construct the canonical Brownian motion on these spaces and use it to solve long-standing questions on diffusions in random media, in particular de Gennes' ``Ant in a Labyrinth'' problem for critical percolation. - Use random planar geometric structures to answer questions about embeddings of metric spaces which are in turn motivated by algorithms used in computer science.
Plain English summaries and category classifications on this site are generated by AI and may not perfectly reflect the original research.
Is something wrong? Let us know