Planar maps with large faces/vertices: an overview

• Nicolas Curien (University Paris-Saclay)

Since the breakthrough works of Le Gall and Miermont, the Brownian sphere has been proved to be the universal limit of many classes of planar map models. One way to escape this paradigm is to consider planar maps models with large degree faces or vertices. We shall survey what we know about the large scale properties of those objects as well as the many questions that remain open at this stage. Based on joint works with Timothy Budd and with Grégory Miermont and Armand Riera.

Quantum Spacetime and the Renormalization Group

• Frank Saueressig (Radboud University)

Functional renormalization group methods are a powerful tool to study the effect of statistical and quantum fluctuations. They also apply to theories including fluctuations in the spacetime metric, where they provide non-trivial evidence for the Reuter fixed point underlying the gravitational asymptotic safety program. Similarly to Wilson-Fisher fixed point visible in three-dimensional scalar field theory, the Reuter fixed point constitutes an non-trivial renormalization group fixed point on the space of theories constructed from the spacetime metric. In this talk, I will give a brief introduction to the functional renormalization group and summarize the key properties of the Reuter fixed point. Special emphasis will be on quantities which potentially lend themselves to a comparison with other computational methods. Current limitations and future perspectives related to the approach will be discussed as well.

Composite operators in Asymptotic Safety

• Carlo Pagani (Johannes Gutenberg-Universität Mainz)

We present how to introduce composite operators in the functional renormalization group formalism and how their scaling properties are retrieved in this approach. We then present the application of this framework to the Asymptotic Safety scenario for quantum gravity and discuss possible avenues to make contact with observable quantities.

Last car decomposition of planar maps

• Alice Contat (University Paris-Saclay)

Consider a tree with n vertices and let $m \geq n$ cars arrive on its vertices. Each car tries to park on its arrival node, and if the spot is already occupied, it drives towards the root of the tree and park as soon as possible. When at the end, all spots are occupied we call the resulting tree a fully parked tree. In this talk, I will describe a new method to enumerate maps that is inspired from Lackner—Panholzer "last car decomposition" of fully parked trees.

Scale-invariant random geometries from mating of trees

• Alicia Castro (Radboud University)

In this talk, I present new results on the search for scale-invariant random geometries in the context of Quantum Gravity. To uncover new universality classes of such geometries, we generalized the mating of trees approach, which encodes Liouville Quantum Gravity on the 2-sphere in terms of a correlated Brownian motion describing a pair of random trees. We extended this approach to higher-dimensional correlated Brownian motions, leading to a family of non-planar random graphs that belong to new universality classes of scale-invariant random geometries. We developed a numerical method to efficiently simulate these random graphs and explore their scaling limits through distance measurements, allowing us to estimate Hausdorff dimensions in the two- and three-dimensional settings.

Enumeration of rectangulations

• Éric Fusy (CNRS, U Marne-la-Vallée)

I will present results for the exact and asymptotic enumeration of generic rectangulations (i.e., tilings of a rectangle by rectangles, with no point where 4 rectangles meet, considered under equivalence relation in a weak or strong form). These can be set in correspondence to models of decorated planar maps (bipolar orientations in the weak form, transversal structures in the strong form) that themselves can be encoded by certain quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson. I will also mention an extension of the results to non-generic rectangulations, which yields a continuum of models of random planar lattices that get closer to a regular lattice.

Statistical physics and geometry of two dimensional materials in classical and quantum cases

• Mikhail Katsnelson (Radboud University)

Discovery of graphene and thermally induced ripples in graphene created a new playground for statistical mechanics of two-dimensional membranes embedded into three-dimensional Euclidean space [1,2]. I will give a general review of the problem including main experimental observations and computer simulation results. After that, I will focus on recent works [3-5] based on the use of the methods of quantum field theory, especially renormalization group. It turns out that the membranes provide a nontrivial example of strongly interacting field theory with scaling invariance but without conformal invariance. At low temperatures the membranes are in quantum regime which is characterized by unusual thermal properties. In particular, thermal expansion coefficient α remains constant till very low temperatures and, instead of vanishing by power-law in temperature T, α ~ Ta as in any “normal” crystals it behaves like α ~ 1/(ln|T|)4/7. At the end I will discuss briefly some open questions such as statistical mechanics of compressed membranes. [1] M. I. Katsnelson and A. Fasolino, Graphene as a prototype crystalline membrane, Acc. Chem. Research 46, 97 (2013) [2] M. I. Katsnelson, The Physics of Graphene (Cambridge Univ. Press, 2020), Chapter 9. [3] A. Mauri and M. I. Katsnelson, Scaling behavior of crystalline membranes: An ε-expansion approach, Nucl. Phys. B 956, 115040 (2020) [4] A. Mauri and M. I. Katsnelson, Scale without conformal invariance in membrane theory, Nucl. Phys. B 969, 115482 (2021) [5] A. Mauri and M. I. Katsnelson, Perturbative renormalization and thermodynamics of quantum crystalline membranes, Phys. Rev. B 105, 195434 (2022)

Conformal removability of SLE$_\kappa$ for $\kappa \in [4,8)$

• Jason Miller (University of Cambridge)

We consider the Schramm-Loewner evolution (SLE$_\kappa$) with $\kappa=4$, the critical value of $\kappa > 0$ at or below which SLE$_\kappa$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable. Such curves arise as the conformal welding of a pair of independent critical ($\gamma=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\mathbf C}$ to be conformally removable which applies in the case that $X$ is not necessarily the boundary of a simply connected domain. We will also describe how this theorem can be applied to obtain the conformal removability of the SLE$_\kappa$ curves for $\kappa \in (4,8)$ in the case that the adjacency graph of connected components of the complement is a.s. connected. Based on joint work with Konstantinos Kavvadias and Lukas Schoug.

Random Disks of Constant Curvature

• Frank Ferrari (Université Libre de Bruxelles)

Two-dimensional quantum gravity models fall in three classes: Liouville gravity, for which the geometry is wildly random in the bulk; topological gravity, for which the geometries, having constant curvature and geodesic boundaries, have a finite number of moduli; and an intermediate class of models, which has attracted a lot of attention recently, for which the metrics have constant curvature but the boundaries can fluctuate wildly. These so-called Jackiw-Teitelboim models have been studied in the physics literature, mainly for negative curvature, under poorly understood assumptions and in a certain limit for which the boundary length goes to infinity. The aim of the talk will be to present a first-principle approach to JT gravity without assuming that a particular limit is taken, mainly focusing on the disk topology. Because the curvature is fixed, the randomness entirely comes from the fluctuating disk boundary, which is a closed curve immersed in the hyperbolic space, the plane or the sphere. The goal is thus to count closed curves that bound a disk, a very interesting and non-trivial combinatorial problem. Our construction, which relies on good old conformal gauge techniques and insights from recent developments, yields a new class of random geometrical models with many properties that remain to be explored.

Measuring hyperbolic surfaces by counting trees

• Bart Zonneveld (Radboud University)

The moduli space of hyperbolic surfaces (roughly the set of all unique hyperbolic surfaces) is an interesting object in random geometry and its properties also have applications in a quantum gravity toy-model called JT gravity. There exists a topological recursion by Mirzakhani which allows us to compute the total volumes of these moduli spaces, but this formulation gives us little insight in the different contributing surfaces. In this talk I will discuss another formulation that allows us to describe the moduli space (at least for genus 0) using trees with simple additional data. This tree bijection reproduces the same total volumes as Mirzakhani’s recursion, but also opens the possibility to look at more complicated statistics.

Gradient squared of the Gaussian free field and the Abelian Sandpile Model: A connection

• Alan Rapoport (Utrecht University)

The discrete Gaussian free field (DGFF) is a famous model in statistical mechanics. In this work we study the scaling limit of a non-linear transformation of its gradient. More precisely, we study the (centered) square of the norm of the gradient DGFF at every point of a square lattice. One of the reasons for studying this object stems from the so-called Abelian sandpile, which is an example of a dynamical system displaying self-organized criticality. Surprisingly, our real-valued model is connected to the height-one field in the sandpile, which only assumes the values 0 or 1. With different methods we are able to obtain the same scaling limits: on the one hand, we show an identity relating the cumulants of our model to those of the height-one field. Besides, we show our field converges to white noise in the limit, as it happens for the height-one field. Joint work with Alessandra Cipriani (UCL), Rajat Subhra Hazra (Leiden) and Wioletta Ruszel (Utrecht).

Problem session

Cycles in permutations and cylinders in square-tiled surfaces

• Vincent Delecroix (LaBRI, Université de Bordeaux)

The limit behavior of cycles in random permutations has attracted a lot of interest. For example it is well known that the number of small cycles follows a Poisson distribution. Similar limit laws as the genus tends to infinity exist for square-tiled surfaces (which are special types of quadrangulations close to the fully packed loop O(1) model). Our result holds in a specific case but numerical experiments suggest that it holds beyond our restricted setting. I also aim to discuss a possible interpolation between permutations and square-tiled surfaces. This is a joint work with E. Goujard, P. Zograf, A. Zorich on the one hand and with M. Liu on the other hand.

Cutting planar maps into slices

• Jérémie Bouttier (IPhT, CEA, University Paris-Saclay)

Maps are discrete surfaces obtained by gluing polygons, and form an important model of 2D random geometry. Among the many approaches developed to study them, the bijective method has been instrumental in understanding their metric properties and their scaling limits. Originally the method consisted in finding bijections between planar maps and certain labeled/decorated trees, called blossom trees or mobiles. It was more recently realized that the recursive structure of trees could be directly implemented at the level of maps, via the so-called "slice decomposition". I will present the main ideas of this method. Based on collaborations with Emmanuel Guitter, Marie Albenque, and Grégory Miermont.

Curvature in Random Geometry

• Renate Loll (Radboud University)

Curvature, as captured by the Riemann curvature tensor, is a key notion in (pseudo-)Riemannian geometry and General Relativity. Nonperturbative quantum gravity, formulated in terms of more general, "random" metric spaces, like causal dynamical triangulations (CDT), forces us to take a new perspective on curvature. Because of the intrinsic coordinate invariance of such approaches, this perspective may in some sense be more physical than that of standard GR textbook folklore, but at the same time raises interesting questions. Do meaningful notions of curvature exist in a highly nonclassical regime? What is more elementary, "the metric" (lengths and angles) or "curvature"? In the absence of tensor calculus, can we capture directional aspects of curvature? How does curvature behave under a change of scale? -- I will summarize some insights that have been gained so far by studying the new Quantum Ricci Curvature in the context DT and CDT quantum gravity.

Universe through Random Geometries

• Dániel Németh (Radboud University)

The study of random geometries is a growing area of research that has broad implications for our understanding of quantum gravity and the nature of the universe. One of the models that give physical meaning to geometries is called Causal Dynamical Triangulations, which is a lattice formulation of gravity. With the help of Monte Carlo simulations, we are allowed to generate and analyze many different geometries, allowing us to explore the properties of spacetime and gravity. The path integral formulation is a key method to dynamically select the relevant geometries from the vast possibilities of all triangulations in a well-defined way. In this talk, I will discuss how to perform numerical simulations to generate random geometries and give rise to meaningful quantities such as topology, dimension, or even a whole universe.

Brownian surfaces

• Jérémie Bettinelli (École polytechnique)

Similarly to Brownian motion, which appears as the universal scaling limit of any reasonable random walk, Brownian surfaces are random metric spaces that appear as the universal scaling limit of reasonable models of random maps of a given surface. These objects generalize the Brownian sphere of Miermont and Le Gall, which is obtained when considering random maps of the sphere. We will present Brownian surfaces and give some of their remarkable properties. This work is in common with Grégory Miermont.