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...

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...

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.

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...

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...

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)...

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...

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)...

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...

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 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...

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...

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...

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...

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...

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...