Critical and supercritical percolation

2026-07-22

Percolation studies how independent random input that is spread uniformly on a lattice or in space gives rise to macroscopic structures. This model, initially introduced to understand porosity, has turned out to be central for understanding fundamental features of real-world phenomena, ranging from phase transitions in physical and chemical systems to stability of Boolean functions with respect to perturbations. Over the last sixty years, a number of important mathematical results have been obtained concerning percolation, with ideas, interactions and consequences in mathematical fields such as probability, combinatorics, complex analysis, geometric group theory, planar topology and theoretical computer science. Highlights include the rigorous derivation of a number of features that are shared with other models from statistical physics: sharpness of phase transitions, renormalization theory, existence of scaling limits and critical exponents, relationship between discrete and continuous descriptions (constructive field theory)... The story is however incomplete, as some of the most fundamental questions have not yet found a mathematical answer. Two notable examples that motivate the present research proposal are the continuity of the phase transition for Bernoulli percolation in dimension three (does the macroscopic structure appear continuously?) and the universality of planar percolation (are the macroscopic features of critical percolation in two dimensions independent of the microscopic model under consideration?). In light of very recent progress, we propose here a list of interrelated projects, with the global aim of developing new tools that should enable us to make progress towards these two open problems. The impact of this study would go beyond the percolation or statistical physics community, as we aim to provide a clean and thorough understanding of some key concepts and phenomena, that would find natural applications in other disciplines.

Switzerland

2026-02-01