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Mini-school on Universality in mathematical physics: random geometries, field theories and hydrodynamics
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Europe/Paris
Salle Condorcet (ENS de Lyon)
Salle Condorcet
ENS de Lyon
1 place de l'École
Description
A central idea in physics is to describe the macroscopic properties of a system with a large number of degrees of freedom by an effective continuous model. Such model is most relevant when it is universal, in the sense that it does not depend on the microscopic details of the system. While conceptually appealing, this approach is often very difficult to implement in a mathematically rigorous way. Still, important progress has been made recently in different directions, such as random geometries emerging from discrete tessellations, non linear dynamics arising in out-of-equilibrium systems, stochasticity appearing spontaneously in the theory of turbulence, etc. The purpose of the mini-school was to present some of these developments, and to allow people working in these diverse topics to exchange ideas.
The program consisted of 5 courses (3 lectures of 90 mn each).
All lectures were held in Salle Condorcet (map link).
Lecturers : Isabelle Gallagher, Ewain Gwynne, Alexei Mailybaev, Daniel Remenik, Slava Rychkov.
Organizers : Oriane Blondel, Jérémie Bouttier, Laurent Chevillard, Christophe Garban, Grégory Miermont.
Support : LABEX MILYON (trimester PhysMathLyon), Simons Collaboration on Wave Turbulence, Laboratoire de Physique.
Description of the courses :
- Isabelle Gallagher : From an atomistic description of gases to fluctuating hydrodynamics
It is well-known that fluid dynamics equations can be obtained rigorously from the Boltzmann equation in a limit when the Knudsen number tends to zero. The derivation of the Boltzmann equation from a deterministic, atomistic description of gases, following Newton's laws, is also known to be possible. Unfortunately this derivation holds on a time interval that shrinks to zero with the Knudsen number, thus preventing to this day any rigorous derivation of fluid dynamics equations from particle systems. In these lectures we shall first explain these different descriptions and the link between them, and then focus on the study of a gas at equilibrium: we shall see that in this setting it is possible to reconcile both descriptions and that fluctuating (linear) hydrodynamic equations can be derived from the deterministic particle system.
This is a joint work with T. Bodineau, L. Saint-Raymond, and S. Simonella. - Ewain Gwynne : The Liouville quantum gravity metric (lecture notes)
Liouville quantum gravity (LQG) is a universal one-parameter family of random fractal surfaces. These surfaces were first introduced (non-rigorously) in the setting of string theory, and are expected to describe the scaling limits of various types of random planar maps.
Recent works have shown that one can endow an LQG surface with a metric (distance function). This metric has many interesting geometric properties. For example, its Hausdorff dimension is strictly greater than 2 and its geodesics merge into each other and form a tree-like structure.
I will discuss the definition of and motivation for LQG, the construction and properties of the metric, and some open problems. I will not assume any background knowledge beyond undergraduate-level probability theory. - Alexei Mailybaev : Renormalization group for spontaneous stochasticity in turbulence
The phenomenon of spontaneous stochasticity in turbulence describes intrinsically stochastic solutions to a deterministic initial value problem, i.e., deterministic scale-invariant equations of motion (such as Euler equations for ideal fluid) and deterministic initial conditions. These solutions arise in the limit of vanishing viscosity and infinitesimal small-scale noise, e.g. thermal fluctuations. We discuss existing evidence of the spontaneous stochasticity in the Eulerian form, i.e., spontaneous randomness of velocity fields. Then, we consider rigorously solvable models. Finally, we study the relation of spontaneous stochasticity with a scaling symmetry, leading to a formulation of the renormalization group theory. - Daniel Remenik : Polynuclear growth and the KPZ fixed point
The KPZ universality class is a broad collection of probabilistic models including one-dimensional random growth, directed polymers and particle systems. At its center lies the KPZ fixed point, a special scaling invariant Markov process which governs the asymptotic fluctuations of all models in the class. The transition probabilities of the KPZ fixed point are known very explicitly, and they satisfy a famous integrable PDE. In these lectures I will discuss these topics using the polynuclear growth model (PNG) as a starting point. This is a model for crystal growth in one dimension, which is intimately connected to the classical longest increasing subsequence problem for a uniformly random permutation. I will explain how PNG can be solved explicitly through probabilistic arguments, and how this solution yields a connection between PNG and another classical completely integrable system. - Slava Rychkov : Tensor renormalization group for lattice models (paper)
Fifty years ago, theoretical physicist Kenneth Wilson conjectured that critical points of lattice models (such as the Ising model) correspond to nontrivial fixed points of a renormalization group transformation. As of now, this is still a widely open problem in mathematics. In these lectures we will discuss related intuitions, results and possible methods of attack. This course was based on the recent paper arXiv:2210.06669 with Tom Kennedy. See also arXiv:2107.11464 and arXiv:2008.04361.
