Here is a brief outline the main achievements in the three directions of the ERC LiKo project.
1) Liouville quantum gravity. Holden, Garban, Sepúlveda (postdoc funded by this ERC) and Sun have investigated dynamical percolation evolving in a Liouville quantum gravity random environment. This work provides a key ergodicity ingredient which was necessary in order to implement the so-called Cardy's embedding program discovered by Holden and Xun. This was one of the major goals of this project. Aru, Powell and Sepúlveda have obtained a remarkable construction of the critical Liouville measures as limits of sub-critical Liouville measures. Their work has identified the appearance of a surprising and subtle factor 2 which does not show up when one were to manipulate double limits without care. Aru, Lupu and Sepúlveda investigated the puzzling fractal geometry of first-passage and two-valued sets of the two dimensional Gaussian Free Field. Chen, Garban and Shekhar (postdoc funded by the ERC) obtained a characterisation of the top particles of a Branching Brownian motion as the unique fixed point of a natural Markov process on point processes. Finally, Garban studied a dynamics which belongs to the broad class of singular stochastic PDEs (which includes KPZ, dynamical Phi^4 etc.) and which is built to preserve the Liouville measure. Interestingly this SPDE exhibits some new features due to the presence of intermittence.
2) Euclidean statistical physics. In this direction, Vanneuville (PhD funded by this ERC) obtained several striking results 1. Together with Rivera and other co-authors, he obtained important results on the random geometry of nodal lignes of a large family of smooth Gaussian fields. 2. Together with Garban, he obtained the existence of exceptional times for conservative dynamics preserving critical percolation. 3. Finally Vanneuville deepened the understanding of critical percolation in a quenched disorder given by Poisson-Voronoi cells. Garban has then used such conservative exclusion processes to answer a question from geometric group theory. Garban and Sepúlveda found a new type of phase transition for the Discrete Gaussian Free Field which is related to the BKT transition of the XY model. This work builds on a seminal work of Fröhlich and Spencer and gives some partial theoretical justification to the conjectured appearance of CLE_4, SLE_4 for the analysis of the XY model at low temperatures. Finally, Duminil-Copin, Garban and Tassion established an unexpected first-order phase transition for the "Book-Ising model" (i.e. several 2d Ising models are glued together along a 1d line). This work was motivated by an intriguing behaviour of the Renyi entropy of certain quantum spin systems.
3)3d turbulence. In this third direction, together with Pereira and Chevillard, Garban has studied (both from theoretical and numerical sides) a stochastic model for 3d turbulence which was suggested by a work of Chevillard, Robert, Vargas in 2010. This stochastic model matches perfectly with velocity fields measured in real turbulent flows. Subsequently, a work by Chevillard, Garban, Rhodes and Vargas strengthened the mathematical analysis of this stochastic model.
