Periodic Reporting for period 3 - CriSP (Critical and supercritical percolation)
Periodo di rendicontazione: 2023-09-01 al 2025-02-28
Over the last sixty years, the probabilistic analysis of models in statistical mechanics has become a dynamic branch of research, with numerous interactions with other fields of mathematics: graph theory, complex analysis, geometric group theory, topology, computer science, to name a few. Initially introduced as a model for porosity, percolation has become fundamental in the theoretical understanding of phase transitions. Developing percolation theory has been a source of several fascinating mathematical challenges, but also has provided a solid mathematical support to key concepts in statistical physics: sharpness of phase transition, renormalization theory, existence of scaling limits and critical exponents, relationship between discrete and continuous descriptions (constructive field theory), universality at criticality, existence of a critical dimension for mean-field behavior... Percolation theory is now mature and well established, but the picture is still not complete: a few basic questions have not found an answer and are now considered as some of the most challenging questions in probability. In particular, two major open problems witness the current lack of understanding in the field: the continuity of the phase transition in dimension three, and the universality in dimension two. In this this project, we aim at developing new tools in order to make progress towards the above mentioned open problems.
Towards the understanding of universality of planar percolation, Russo-Seymour-Welsh theory plays a central role. Originally developed for Bernoulli percolation in the seventies, it has recently been extended to other models (eg FK percolation, level lines of random fields) leading to major breakthroughs. The PI and Laurin Köhler-Schindler (PhD student funded by CRISP) proved a general Russo-Seymour-Welsh result for positively associated measures. This unifies all the previous proofs, but also provides a general tool for the study of critical percolation systems with positive correlations.
Noise sensitivity of Boolean functions has been the object of intense studies in the 2000's. The results have several applications, such as the study of exceptional times for dynamical percolation, or quantitative correlation inequalities. Most of the theory relies on Fourier methods, which makes it restrictive to product measures. The PI and Hugo Vanneuville developed a geometric approach to noise sensitivity, which does not rely on Fourier arguments. The method is more robust and already lead to applications to noise sensitivity for Glauber dynamics. We also believe that this will provide some important tools towards the main questions in our proposal (3D critical percolation, 2D universality).
In the last years, numerous breakthroughs have improved the understanding of the sharpness phenomena in percolation theory. Our group has been very active in this area. A central work was written by the PI together with Hugo Duminil-Copin and Aran Raoufi, developing a method based on randomized algorithms. The method has been successfully applied to several processes. For instance, Franco Severo (a postdoc funded by CRISP) together with Hugo Duminil-Copin, Subajit Goswami and Pierre-François Rodriguez obtained sharpness for Gaussian percolation, which was a major conjecture in the field. In another direction, The PI and Barbara Dembin (postdoc funded by CRISP) obtained a sharpness result for Boolean percolation.
Noise sensitivity of Boolean functions has been the object of intense studies in the 2000's. The results have several applications, such as the study of exceptional times for dynamical percolation, or quantitative correlation inequalities. Most of the theory relies on Fourier methods, which makes it restrictive to product measures. The PI and Hugo Vanneuville developed a geometric approach to noise sensitivity, which does not rely on Fourier arguments. The method is more robust and already lead to applications to noise sensitivity for Glauber dynamics. We also believe that this will provide some important tools towards the main questions in our proposal (3D critical percolation, 2D universality).
In the last years, numerous breakthroughs have improved the understanding of the sharpness phenomena in percolation theory. Our group has been very active in this area. A central work was written by the PI together with Hugo Duminil-Copin and Aran Raoufi, developing a method based on randomized algorithms. The method has been successfully applied to several processes. For instance, Franco Severo (a postdoc funded by CRISP) together with Hugo Duminil-Copin, Subajit Goswami and Pierre-François Rodriguez obtained sharpness for Gaussian percolation, which was a major conjecture in the field. In another direction, The PI and Barbara Dembin (postdoc funded by CRISP) obtained a sharpness result for Boolean percolation.
Based on our progress in the first part of the project, we are planning to work mainly in the following four directions:
- Noise sensitivity of percolation for Glauber dynamics.
- Robust approaches to supercritical percolation (in particular, the study of Wulff shapes).
- A geometric interpretation of 2D correlation inequalities.
- 3D critical percolation.
- Noise sensitivity of percolation for Glauber dynamics.
- Robust approaches to supercritical percolation (in particular, the study of Wulff shapes).
- A geometric interpretation of 2D correlation inequalities.
- 3D critical percolation.