## Periodic Reporting for period 3 - LiKo (From Liouville to Kolmogorov: 2d quantum gravity, noise sensitivity and turbulent flows)

Reporting period: 2019-09-01 to 2021-02-28

This research project is organized along three seemingly unrelated directions:

(1) Mathematical Liouville gravity deals with the geometry of large random planar maps. Historically, conformal invariance was a key ingredient in the construction of Liouville gravity in the physics literature. Conformal invariance has been restored recently with an attempt of understanding large random combinatorial planar maps once conformally embedded in the plane. The geometry induced by these embeddings is conjecturally described by the exponential of a very oscillating distribution, the Gaussian Free Field. This conjecture is part of a broader program aimed at rigorously understanding the celebrated KPZ relation. The first major goal of this project is to make progress towards the completion of this program. Several tools such as Liouville Brownian motion, circle packings, QLE processes and Bouchaud trap models will be combined.

(2) Euclidean statistical physics is closely related to area (1) through the above KPZ relation. The plan is to push further the analysis of critical statistical physics models successfully initiated by the works of Schramm and Smirnov. I will focus in particular on dynamics at and near critical points with a special emphasis on the so-called noise sensitivity of these systems.

(3) 3d turbulence. A more tractable ambition than solving Navier-Stokes equation is to construct explicit stochastic vector fields which combine key features of experimentally observed velocity fields. I will make the mathematical framework precise by identifying four axioms that need to be satisfied. It has been observed recently that the exponential of a certain log-correlated field, as in (1), could be used to create such a realistic velocity field. The main objective here is to construct and analyse this challenging object by relying on techniques from (1) and (2). This would be the first genuine stochastic model of turbulent flow in the spirit of what Kolmogorov was aiming at.

(1) Mathematical Liouville gravity deals with the geometry of large random planar maps. Historically, conformal invariance was a key ingredient in the construction of Liouville gravity in the physics literature. Conformal invariance has been restored recently with an attempt of understanding large random combinatorial planar maps once conformally embedded in the plane. The geometry induced by these embeddings is conjecturally described by the exponential of a very oscillating distribution, the Gaussian Free Field. This conjecture is part of a broader program aimed at rigorously understanding the celebrated KPZ relation. The first major goal of this project is to make progress towards the completion of this program. Several tools such as Liouville Brownian motion, circle packings, QLE processes and Bouchaud trap models will be combined.

(2) Euclidean statistical physics is closely related to area (1) through the above KPZ relation. The plan is to push further the analysis of critical statistical physics models successfully initiated by the works of Schramm and Smirnov. I will focus in particular on dynamics at and near critical points with a special emphasis on the so-called noise sensitivity of these systems.

(3) 3d turbulence. A more tractable ambition than solving Navier-Stokes equation is to construct explicit stochastic vector fields which combine key features of experimentally observed velocity fields. I will make the mathematical framework precise by identifying four axioms that need to be satisfied. It has been observed recently that the exponential of a certain log-correlated field, as in (1), could be used to create such a realistic velocity field. The main objective here is to construct and analyse this challenging object by relying on techniques from (1) and (2). This would be the first genuine stochastic model of turbulent flow in the spirit of what Kolmogorov was aiming at.

In each of the three directions listed above, some significant progress has been made since the beginning of this project. Let me briefly outline the main achievements so far in each direction.

1) Liouville quantum gravity. Avelio Sepulveda (postdoc funded by this ERC) has obtained remarquable results together Juhan Aru and Ellen Powell. They managed in particular to build the so called 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. Together with Juhan Aru, Avelio Sepulveda also investigated the puzzling fractal geometry of first-passage and two-valued sets of the two dimensional Gaussian Free Field. Together with Nina Holden and Xin Sun, Christophe Garban and Avelio Sepulveda have started investigating dynamical percolation evolving in a Liouville quantum gravity random environment. During the period covered by the report, they obtained a control of the atypical behaviour of Gaussian multiplicative Chaos, when integrated against fractal measures (instead of more standard diffuse measures). This work will be used in the subsequent work by the same authors whose aim is to provide the main technical tool in order to implement the so-called Cardy's embedding. This is an explicit way of conformally embedding discrete maps discovered by Nina Holden and Xin Sun. Our work will ultimately imply the convergence of this Cardy's embedding to the Liouville measure which was one of the major goals of this project. Finally, Christophe Garban has 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. Along this direction, Hugo Vanneuville (PhD funded by this ERC) obtained striking results in three directions. 1. Together with Alejandro Rivera and other co-authors, he obtained important results on the random geometry of random nodal lignes of a large family of smooth Gaussian fields. Building on earlier works of Beffara-Gayet, he described with Alejandro Rivera the phase transition for level sets of these Gaussian fields. 2. Together with Christophe Garban, he obtained the existence of exceptional times for conservative dynamics preserving critical percolation in 2d (made of exclusion processes). The analysis required a subtle analysis of how Fourier coefficients of percolation diffuse under exclusion processes. 3. Finally he has deepened the understanding of critical percolation in random scenery provided by Poisson-Voronoi cells. In a different direction, Christophe Garban has used such conservative exclusion processes in order to answer a question from geometric group theory.

3)3d turbulence. In this third topic, together with Rodrigo Pereira and Laurent Chevillard, Christophe 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 (appropriately modified in order to make the analysis more under reach). We thus constructed, for the first time to our knowledge, a stochastic model which combines the main features of turbulent flows (negative skewness, intermittence, etc.)

1) Liouville quantum gravity. Avelio Sepulveda (postdoc funded by this ERC) has obtained remarquable results together Juhan Aru and Ellen Powell. They managed in particular to build the so called 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. Together with Juhan Aru, Avelio Sepulveda also investigated the puzzling fractal geometry of first-passage and two-valued sets of the two dimensional Gaussian Free Field. Together with Nina Holden and Xin Sun, Christophe Garban and Avelio Sepulveda have started investigating dynamical percolation evolving in a Liouville quantum gravity random environment. During the period covered by the report, they obtained a control of the atypical behaviour of Gaussian multiplicative Chaos, when integrated against fractal measures (instead of more standard diffuse measures). This work will be used in the subsequent work by the same authors whose aim is to provide the main technical tool in order to implement the so-called Cardy's embedding. This is an explicit way of conformally embedding discrete maps discovered by Nina Holden and Xin Sun. Our work will ultimately imply the convergence of this Cardy's embedding to the Liouville measure which was one of the major goals of this project. Finally, Christophe Garban has 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. Along this direction, Hugo Vanneuville (PhD funded by this ERC) obtained striking results in three directions. 1. Together with Alejandro Rivera and other co-authors, he obtained important results on the random geometry of random nodal lignes of a large family of smooth Gaussian fields. Building on earlier works of Beffara-Gayet, he described with Alejandro Rivera the phase transition for level sets of these Gaussian fields. 2. Together with Christophe Garban, he obtained the existence of exceptional times for conservative dynamics preserving critical percolation in 2d (made of exclusion processes). The analysis required a subtle analysis of how Fourier coefficients of percolation diffuse under exclusion processes. 3. Finally he has deepened the understanding of critical percolation in random scenery provided by Poisson-Voronoi cells. In a different direction, Christophe Garban has used such conservative exclusion processes in order to answer a question from geometric group theory.

3)3d turbulence. In this third topic, together with Rodrigo Pereira and Laurent Chevillard, Christophe 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 (appropriately modified in order to make the analysis more under reach). We thus constructed, for the first time to our knowledge, a stochastic model which combines the main features of turbulent flows (negative skewness, intermittence, etc.)

In the remaining part of this project, we will be aiming at pushing each of these three directions further (as initially planned in the proposal). More precisely:

1) On the Liouville quantum gravity side, we will finish our above mentioned ongoing project which involves Christophe Garban, Nina Holden, Avelio Sepulveda, Xin Sun and whose purpose is to analyze the ergodicity properties of dynamical Liouville percolation. We also plan (together with Atul Shekhar who sill start his postdoc next september) to pursue the analysis of the Liouville SPDE.

2) On the Euclidean statistical physics side, the plan is next to analyze the noise sensitivity of percolation models induced by smooth Gaussien Fields (such as the one studied by Hugo Vanneuville, PhD of the project), and to push further the understanding of conservative dynamics. New directions of interest in this topic have also been identified and will be analysed further.

3) 3d turbulence. With the paper involving Chevillard, Garban, Rhodes, Vargas, the main goal of the third part of this project has now achieved but only in dimension 1 so far. (As was initially planned in the project). It therefore remains to extend this stochastic model to a homogeneous, incompressible stochastic vector field on R^3.

1) On the Liouville quantum gravity side, we will finish our above mentioned ongoing project which involves Christophe Garban, Nina Holden, Avelio Sepulveda, Xin Sun and whose purpose is to analyze the ergodicity properties of dynamical Liouville percolation. We also plan (together with Atul Shekhar who sill start his postdoc next september) to pursue the analysis of the Liouville SPDE.

2) On the Euclidean statistical physics side, the plan is next to analyze the noise sensitivity of percolation models induced by smooth Gaussien Fields (such as the one studied by Hugo Vanneuville, PhD of the project), and to push further the understanding of conservative dynamics. New directions of interest in this topic have also been identified and will be analysed further.

3) 3d turbulence. With the paper involving Chevillard, Garban, Rhodes, Vargas, the main goal of the third part of this project has now achieved but only in dimension 1 so far. (As was initially planned in the project). It therefore remains to extend this stochastic model to a homogeneous, incompressible stochastic vector field on R^3.