Final Report Summary - CONFRA (Conformal fractals in analysis, dynamics, physics)
During the last two decades conformally invariant fractal structures emerged in several areas of mathematics and physics. The goal of the CONFRA project was to study such structures arising in analysis, dynamics, probability, physics, and exploit interaction of these fields. The emphasis was on conformal invariance emerging in the scaling limits of 2D lattice models of statistical physics at criticality, such as the percolation model (for movement of liquid through a porous medium), the Ising model of a ferromagnet, or the self-avoiding walk as a model for a polymer molecule.
Among the main achievements are:
Universality and conformal invariance in the Ising model
Ising model provides an archetypical example of an order-disorder phase transition, and was extensively studied by physicists and mathematicians alike. Conformal invariance and universality of its scaling limit at criticality were widely accepted and allowed unrigorous derivation of many properties, however they were never established mathematically. Chelkak and Smirnov have established that the critical Ising model on a large family of planar graphs has a conformally invariant universal scaling limit, thus proving the old belief, and opening the way to rigorous determination of many dimensional and scaling properties.
The relevant graphs, the so-called rhombic lattices and isoradial graphs, form the largest family where the Ising critical point has been derived, so this essentially establishes the universality conjecture for the Ising model at criticality.
Description of the full conformally invariant scaling limits
Schramm and Smirnov have shown that any (subsequential) scaling limit of standard 2D percolation models at criticality can be described as a noise, or a continuous product of probability spaces. The resulting noise is non-classical, or black in the terminology of Tsirelson, thus providing the first example of a 2D black noise, a fundamental object. Also a new setup is provided for describing scaling limits of 2D cluster models in terms of connectivity properties.
Derivation of the asymptotic number of the self-avoiding walks on the hexagonal lattice
Duminil-Copin and Smirnov derived the value of the connective constant of the hexagonal lattice, predicted by physicist Nienhuis. The works paves way to study further properties of the self-avoiding walks.
A new approach to massive models and field theories
Makarov and Smirnov developed an approach to describe the scaling limits of the massive perturbations of the critical lattice models, based on perturbations of holomorphic martingale observables, which satisfy massive versions of Cauchy-Riemann equations. It allows to connect the scaling limits of interfaces to massive versions of Schramm's SLE curves and study properties of the latter.
Among the main achievements are:
Universality and conformal invariance in the Ising model
Ising model provides an archetypical example of an order-disorder phase transition, and was extensively studied by physicists and mathematicians alike. Conformal invariance and universality of its scaling limit at criticality were widely accepted and allowed unrigorous derivation of many properties, however they were never established mathematically. Chelkak and Smirnov have established that the critical Ising model on a large family of planar graphs has a conformally invariant universal scaling limit, thus proving the old belief, and opening the way to rigorous determination of many dimensional and scaling properties.
The relevant graphs, the so-called rhombic lattices and isoradial graphs, form the largest family where the Ising critical point has been derived, so this essentially establishes the universality conjecture for the Ising model at criticality.
Description of the full conformally invariant scaling limits
Schramm and Smirnov have shown that any (subsequential) scaling limit of standard 2D percolation models at criticality can be described as a noise, or a continuous product of probability spaces. The resulting noise is non-classical, or black in the terminology of Tsirelson, thus providing the first example of a 2D black noise, a fundamental object. Also a new setup is provided for describing scaling limits of 2D cluster models in terms of connectivity properties.
Derivation of the asymptotic number of the self-avoiding walks on the hexagonal lattice
Duminil-Copin and Smirnov derived the value of the connective constant of the hexagonal lattice, predicted by physicist Nienhuis. The works paves way to study further properties of the self-avoiding walks.
A new approach to massive models and field theories
Makarov and Smirnov developed an approach to describe the scaling limits of the massive perturbations of the critical lattice models, based on perturbations of holomorphic martingale observables, which satisfy massive versions of Cauchy-Riemann equations. It allows to connect the scaling limits of interfaces to massive versions of Schramm's SLE curves and study properties of the latter.