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.
