From Liouville to Kolmogorov: 2d quantum gravity, noise sensitivity and turbulent flows

Project Information

LiKo

Grant agreement ID: 676999

DOI

10.3030/676999

Project closed

EC signature date 1 March 2016

Start date 1 September 2016

End date 31 August 2022

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 935 000,00

EU contribution

€ 935 000,00

Coordinated by

UNIVERSITE LYON 1 CLAUDE BERNARD
France

Periodic Reporting for period 4 - LiKo (From Liouville to Kolmogorov: 2d quantum gravity, noise sensitivity and turbulent flows)

Reporting period: 2021-03-01 to 2022-08-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The main achievements beyond the state of the art in the three main scientific directions of the LiKo project are the following ones:

1) On the Liouville quantum gravity side, the main achievements are 1a) the joint works of Christophe Garban, Nina Holden, Avelio Sepulveda and Xin Sun on the ergodicity properties of dynamical Liouville percolation which contributed to the completion of the Cardy's embedding program by Holden and Sun. 1b) Another major achievement is the series of papers by Juhan Aru, Titus Lupu and Avelio Sepúlveda (Postdoc of the LiKo project) on the puzzling fractal properties of the first and two-valued sets of the continuum Gaussian Free Field which created since then an intense activity. 1c) The analysis of the Liouville SPDE by Christophe Garban also generated significant scientific activity over the last three years within singular Stochastic PDEs.

2) On the Euclidean statistical physics side, the main achievements are as follows 2a) Noise sensitivity of critical percolation under conservative dynamics by Christophe Garban and Hugo Vanneuville (PhD of the LiKo project). 2b) The analysis of critical percolation in a quenched disorder given by the 2d Poisson-Voronoi cells by Hugo Vanneuville. 2c) The statistical reconstruction phase transition for the Discrete Gaussian Free Field which gives a new Berezinsk-Kosterlitz-Thouless type phase transition (by Christophe Garban and Avelio Sepúlveda). 2d) The intriguing first-order phase transition occurring for the "Book-Ising" (by Hugo Duminil-Copin, Christophe Garban and Vincent Tassion). 2e) A new proof of Long-range order for spin systems with continuous symmetry "on the Nishimori line" by Christophe Garban and Thomas Spencer.

3) 3d turbulence. Our main achievement beyond the state of the art in this direction is the joint work by Chevillard, Garban, Rhodes and Vargas which constructed, for the first time to our knowledge, a stochastic model which combines all the main features of turbulent flows (negative skewness, intermittence, etc.)

Long-range order on the "book-Ising"