Periodic Reporting for period 1 - Vortex (Spin systems with discrete and continuous symmetry: topological defects, Bayesian statistics, quenched disorder and random fields)
Okres sprawozdawczy: 2023-09-01 do 2026-02-28
The main goal of this project is to understand the geometry of the very influential topological phase transitions which were discovered in the 70's by Berezinskii, Kosterlitz and Thouless. The archetypal example of such phase transitions arises in the 2d XY model in which topological defects, called vortices, behave very differently at small and high temperature.
The mathematical understanding of this rich phenomenon goes back to the work of Fröhlich and Spencer in the 80's and involves the 2d Coulomb gas. This project is aimed at analyzing this phase transition through the prism of random fractal geometry by associating natural percolating sets to the XY model whose behaviour will depend crucially on the temperature. One constant source of inspiration will be the deep geometric content and powerful probabilistic methods gathered over the last 20 years for celebrated discrete symmetry models such as 2d critical Ising or percolation. New tools will be brought in, among which the recent works of the PI with Sepúlveda which analyze the 2d Coulomb gas and make connections with Bayesian statistics.
Since the early days of topological phase transitions, topological defects have been found to arise also in some discrete symmetry spin systems as well as in Abelian lattice gauge theory in 4d. This project will explore the geometry of these by making several novel and fruitful connections with spins systems such as the Ising model.
The new connections made with statistical reconstruction and Bayesian statistics will give access to the more fascinating and least understood world of spin systems with non-Abelian (gauge-)symmetry.
Finally, we shall investigate the mechanisms which relate the microscopic background noise with the large scale structures it induces in the contexts of Quantum Field Theory and KPZ fixed point.
The impact of this project will go well beyond the current understanding of topological phase transitions in a wide variety of settings where they arise.
The mathematical understanding of this rich phenomenon goes back to the work of Fröhlich and Spencer in the 80's and involves the 2d Coulomb gas. This project is aimed at analyzing this phase transition through the prism of random fractal geometry by associating natural percolating sets to the XY model whose behaviour will depend crucially on the temperature. One constant source of inspiration will be the deep geometric content and powerful probabilistic methods gathered over the last 20 years for celebrated discrete symmetry models such as 2d critical Ising or percolation. New tools will be brought in, among which the recent works of the PI with Sepúlveda which analyze the 2d Coulomb gas and make connections with Bayesian statistics.
Since the early days of topological phase transitions, topological defects have been found to arise also in some discrete symmetry spin systems as well as in Abelian lattice gauge theory in 4d. This project will explore the geometry of these by making several novel and fruitful connections with spins systems such as the Ising model.
The new connections made with statistical reconstruction and Bayesian statistics will give access to the more fascinating and least understood world of spin systems with non-Abelian (gauge-)symmetry.
Finally, we shall investigate the mechanisms which relate the microscopic background noise with the large scale structures it induces in the contexts of Quantum Field Theory and KPZ fixed point.
The impact of this project will go well beyond the current understanding of topological phase transitions in a wide variety of settings where they arise.
Here is a brief outline of the main achievements over the first two years of the ERC project Vortex:
1) WP1 (Topological phase transitions)
The main achievement in this direction is the work by the PI on the invisibility of the integers for the discrete Gaussian Chain. This work analyses to which extend vortices conditioned to leave on a 1D line in Z^2 affect the macroscopic fluctuations. It is shown that in a certain regime, there is not enough room for these vortices to affect the fluctuations (invisibility of the integers) while in another part of the phase diagram it is known since Frohlich and Zegarlinski that such vortices contribute very significantly to the fluctuations (to the point they lead to the localisation of the dual surface). Another achievement is the work by Chevyrev and the PI on Villain's extensions to Amenable and non-Amenable lattice gauge theories.
2) WP2 (Discrete symmetry spin systems)
The main achievements are the work by Easo, Tassion and Severo (Postdoc of the project) "Counting minimal cutsets and $p_c<1$. This is an important paper which now allows to implement a "Peierls type" arguments on a very large family of graphs (way beyond Z^d). Since Peierls argument is the main tool at disposal when analysing discrete symmetry spin systems, this is likely to become a very effective tool when analysing such systems on very general gaphs. Another fruitful line of research are the two works by van Engelenburg (Postdoc of the project), the PI of the project, Panis and Severo (Postdoc of the project), where critical exponents of the Ising and Bernoulli percolation models are obtained. It is shown that one may consider several natural notions of 1-arm exponents and they all differ for Ising model in large enough dimensions. One of the most surprising outcome of these works is the proof that the upper critical dimension of spin Ising is different from the upper-critical dimension of FK-Ising (shown to be 6) despite the existence of very simple "probabilistic bijections" between both models.
3) WP3 (Statistical reconstruction, quenched disorder and non-Abelian continuous symmetry)
The first main achievement in this direction is the work by Aru, the PI and Sepulveda (team member of this project) about the existence of a quenched disorder for the XY model which is compatible with the celebrated prediction of Polyakov of positive mass for the S^2 spin O(3) model on Z^2 at all positive temperatures. This settles a debate which had been popularized by Patrascioiu-Seiler : indeed they argued that Polyakov's prediction was in some sense in contradiction with the possibility of XY to still exhibit BKT power law phase in the presence of disorder. We show in this work that, surprisingly, BKT may fail in a field of very high conductances with very small islands of low conductances. Another achievement in this direction of the project is the work by Paul Dario and the PI which establishes that if the disorder is "iid", for example if the XY model is considered on a supercritical percolation cluster, then the BKT phase still holds at low temperature. Finally, Korzhenkova and Sepulveda (team member of the project) analyse what happens when a N-component GFF is conditioned to avoid a Ball, their analysis is very much motivated by the spin O(N) model and sheds some new light on it.
4) WP4 (Random fields and QFT)
In this direction, the main achievements are the work by the PI and Kupiainen which considers several natural fields in QFT (such as the Energy field of Ising, the so-called Sine-Gordon and the Phi^4 field) and proves that these QFT measures are singular w.r.t. to the non perturbed critical one. This implies in particular that the Energy field of Ising does not exist as a random Schwarz distribution on the plane thus settling an important question in the field. The main idea is to extract singularity of the measures from the mesoscopic inspection of the fields uniformly as the ultraviolet mesh goes to zero.
Another significant achievement is the paper "The supercritical phase of the \varphi^4 model is well behaved" by Franco Severo (Postdoc of the project) and his co-authors where they analyse what happens above the critical beta_c for the lattice Phi^4 model, one of the most important lattice approximation of QFT fields.
1) WP1 (Topological phase transitions)
The main achievement in this direction is the work by the PI on the invisibility of the integers for the discrete Gaussian Chain. This work analyses to which extend vortices conditioned to leave on a 1D line in Z^2 affect the macroscopic fluctuations. It is shown that in a certain regime, there is not enough room for these vortices to affect the fluctuations (invisibility of the integers) while in another part of the phase diagram it is known since Frohlich and Zegarlinski that such vortices contribute very significantly to the fluctuations (to the point they lead to the localisation of the dual surface). Another achievement is the work by Chevyrev and the PI on Villain's extensions to Amenable and non-Amenable lattice gauge theories.
2) WP2 (Discrete symmetry spin systems)
The main achievements are the work by Easo, Tassion and Severo (Postdoc of the project) "Counting minimal cutsets and $p_c<1$. This is an important paper which now allows to implement a "Peierls type" arguments on a very large family of graphs (way beyond Z^d). Since Peierls argument is the main tool at disposal when analysing discrete symmetry spin systems, this is likely to become a very effective tool when analysing such systems on very general gaphs. Another fruitful line of research are the two works by van Engelenburg (Postdoc of the project), the PI of the project, Panis and Severo (Postdoc of the project), where critical exponents of the Ising and Bernoulli percolation models are obtained. It is shown that one may consider several natural notions of 1-arm exponents and they all differ for Ising model in large enough dimensions. One of the most surprising outcome of these works is the proof that the upper critical dimension of spin Ising is different from the upper-critical dimension of FK-Ising (shown to be 6) despite the existence of very simple "probabilistic bijections" between both models.
3) WP3 (Statistical reconstruction, quenched disorder and non-Abelian continuous symmetry)
The first main achievement in this direction is the work by Aru, the PI and Sepulveda (team member of this project) about the existence of a quenched disorder for the XY model which is compatible with the celebrated prediction of Polyakov of positive mass for the S^2 spin O(3) model on Z^2 at all positive temperatures. This settles a debate which had been popularized by Patrascioiu-Seiler : indeed they argued that Polyakov's prediction was in some sense in contradiction with the possibility of XY to still exhibit BKT power law phase in the presence of disorder. We show in this work that, surprisingly, BKT may fail in a field of very high conductances with very small islands of low conductances. Another achievement in this direction of the project is the work by Paul Dario and the PI which establishes that if the disorder is "iid", for example if the XY model is considered on a supercritical percolation cluster, then the BKT phase still holds at low temperature. Finally, Korzhenkova and Sepulveda (team member of the project) analyse what happens when a N-component GFF is conditioned to avoid a Ball, their analysis is very much motivated by the spin O(N) model and sheds some new light on it.
4) WP4 (Random fields and QFT)
In this direction, the main achievements are the work by the PI and Kupiainen which considers several natural fields in QFT (such as the Energy field of Ising, the so-called Sine-Gordon and the Phi^4 field) and proves that these QFT measures are singular w.r.t. to the non perturbed critical one. This implies in particular that the Energy field of Ising does not exist as a random Schwarz distribution on the plane thus settling an important question in the field. The main idea is to extract singularity of the measures from the mesoscopic inspection of the fields uniformly as the ultraviolet mesh goes to zero.
Another significant achievement is the paper "The supercritical phase of the \varphi^4 model is well behaved" by Franco Severo (Postdoc of the project) and his co-authors where they analyse what happens above the critical beta_c for the lattice Phi^4 model, one of the most important lattice approximation of QFT fields.
The main results beyond the state of the arts which have been obtained at this stage of the project are (see previous section for a more detailed account on each of these) :
a) a negative answer to Patrascioiu-Seiler objection to Polyakov's 1975 prediction (Aru, the PI and Sepulveda, team member)
b) a proof that FK-Ising and spin-Ising do not have the same upper critical dimension (van Engelenburg -- Vortex postdoc, the PI, Panis, Severo -- Vortex postdoc).
c) the identification of several critical exponent for critical Ising model in high d, in particular one non-mean field exponent is computed in 3
e) a proof that the Energy field of critical Ising model does not exist as a random schwarz distribution on R^2 (the PI and Kupiainen).
f) A zone with "invisibility" of the integers in models (discrete Gaussian chains) for which visibility arises at both sides of the phase diagram (work by the PI).
g) a new geometric peierls type argument available for very general graphs (Easo, Tassion, Severo -- postdoc of the project).
a) a negative answer to Patrascioiu-Seiler objection to Polyakov's 1975 prediction (Aru, the PI and Sepulveda, team member)
b) a proof that FK-Ising and spin-Ising do not have the same upper critical dimension (van Engelenburg -- Vortex postdoc, the PI, Panis, Severo -- Vortex postdoc).
c) the identification of several critical exponent for critical Ising model in high d, in particular one non-mean field exponent is computed in 3
e) a proof that the Energy field of critical Ising model does not exist as a random schwarz distribution on R^2 (the PI and Kupiainen).
f) A zone with "invisibility" of the integers in models (discrete Gaussian chains) for which visibility arises at both sides of the phase diagram (work by the PI).
g) a new geometric peierls type argument available for very general graphs (Easo, Tassion, Severo -- postdoc of the project).