Programme

Models of two-dimensional random geometry are obtained as universal scaling limits in the Gromov-Hausdorff sense of large graphs embedded in the sphere. These models, which include the Brownian sphere, the Brownian disk and the Brownian plane, are also closely related to the quantum surfaces studied by Miller and Sheffield. We will present recent progress in the study of these random metric spaces. In particular we will discuss some remarkable properties of geodesics and mention some open problems.

Let Tn be a uniformly random tree with vertex set [n]={1, …, n}. Let Δn be the largest vertex degree in Tn and let λn be the largest eigenvalue of Tn. We show that | λn-√Δn| → 0 in probability as n → 0. The key ingredients of our proof are (a) the trace method, (b) a rewiring lemma that allows us to “clean up” our tree without decreasing its top eigenvalue, and (c) some careful combinatorial arguments. 

This is joint work with Roberto Imbuzeiro Oliveira and Gabor Lugosi.

We will be interested in uniform random triangulations with fixed size and genus when the genus is proportional to the size. In this regime, typical triangulations enjoy expander properties which brings them close to random graph models. For example, we will see that graph distances become logarithmic in the size. This contrasts sharply with the fractal nature of random planar (i.e. with genus 0) triangulations. Joint works with Baptiste Louf and Guillaume Chapuy.

We prove that any globally centred discrete snake on a Bienaymé tree with critical, finite third moment offspring distribution which satisfies a finite global variance condition plus a natural tail condition on the displacements has the Brownian snake driven by a Brownian excursion as its scaling limit.  This improves on earlier results of various authors including Janson and Marckert for the case where the displacements are independent of the offspring numbers, and Marckert for the globally centred, global finite variance case restricted to bounded offspring distributions. Our results imply, in particular, the convergence of a special discrete snake which gives the difference between the height process of such a Bienaymé tree and a constant multiple of its Łukasiewicz path, when rescaled by n^{-1/4}, under optimal tail conditions.  Our proof of the convergence of the finite dimensional distributions makes essential use of a discrete line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin; the tightness proof adapts a method deployed by Haas and Miermont in the context of Markov branching trees.

We show that the CRT appears as the limit of random critical trees where deaths and births might be locally correlated. Interestingly, the proof method does not use the classical contour functions but rather the stochastic flow point of view.

I will explain the (classical) link between maps, Hurwitz numbers, and factorisations in the symmetric group, and mention some of the important consequences (Schur function expansions, KP hierarchy) as well as some recent developments and questions (b-deformations, Macdonald deformations). This is meant as an overview talk rather than a talk on precise recent results of mine. If time permits, I will say a word about the strong analogy between maps and Tamari intervals which is still an algebraic mystery/coincidence.

We will discuss a model of random spanning trees obtained by running Wilson's algorithm or the Aldous-Bröder algorithm on the complete graph but replacing the simple random walk with a choice random walk. This leads to a new model of random spanning trees for which the scaling limits can be constructed using stick-breaking constructions of Curien and Haas (2017). The scaling exponents are different to those of uniform spanning trees. Based on joint work with Matan Shalev.

We study a model of random binary trees grown "by the leaves" in the style of Luczak and Winkler (2004). If 𝝉n is a uniform plane binary tree of size n, Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure νn such that the tree obtained by adding a cherry on a leaf sampled according to νn is still uniformly distributed on the set of all plane binary trees with size n+1. It turns out that the measure νn, which we call the leaf-growth measure, is noticeably different from the uniform measure on the leaves of the tree 𝝉n. In fact we prove that as n→∞, with high probability it is almost entirely supported by a subset of only n3(2–√3)+o(1) ≈ n0.8038... leaves.

In the continuous setting, we construct the scaling limit of this measure, which is a probability measure on the Brownian Continuum Random Tree supported by a fractal set of dimension 6(2–√3). We also compute the full (discrete) multifractal spectrum. This work is a first step towards understanding the diffusion limit of the discrete leaf-growth procedure.

Many bijections between maps and decorated trees have been developed in the last 20 years. In 2010, Jérémie Bouttier and Emmanuel Guitter introduced a new bijective paradigm for maps, called the « slice decomposition », which consists in cutting maps along some geodesic paths to produce some sort of canonical building blocks. This decomposition enables to obtain recursive decompositions, similar to the ones already available for decorated trees, but it also leads to new constructions and decompositions. 

In my talk, I will present the extension of the slice decomposition to hypermaps (i.e maps in which faces can be properly coloured in two colours), which permits to derive bijective proofs for enumerative formulas obtained in the physics literature. This is a joint work with Jérémie Bouttier.