Fractal properties of the Aldous-Kendall random metric

• Guillaume Blanc

Room: Salle 435 We consider the random metric constructed by Kendall on $\mathbb{R}^d$ from a self-similar Poisson process of roads, i.e. lines with a speed limit. Intuitively, the process generates a random road network in $\mathbb{R}^d$ that one can travel on, respecting the speed limits; and this induces a random metric $T$ on $\mathbb{R}^d$, for which the distance between points is given by the optimal connection time. In this talk, I will present some fractal properties of the random metric space $\left(\mathbb{R}^d,T\right)$. In particular, although it is almost surely homeomorphic to the usual Euclidean $\mathbb{R}^d$, its Hausdorff dimension is given by $(\gamma-1)d/(\gamma-d)$, where $\gamma>d$ is a parameter of the model. This fractal property, which is reminiscent of the Brownian sphere, confirms a conjecture of Kahn. If time allows, I will also mention some multifractal properties of the metric space $\left(\mathbb{R}^d,T\right)$ equipped with the Lebesgue measure, which in particular distinguish it from the Brownian sphere equipped with its volume measure.

Topological recursion for constellations with internal faces, and tau functions

• Guillaume Chapuy

Room: Salle 435 In enumeration of maps, we are used to universality: if a property is true for, say, quadrangulations (algebraic generating function, nice parametrization with trees, certain counting exponents, ...) an analogue should be true for many other models: 2k-angulations, triangulations, Boltzmann maps, $r$-constellations, ... but where to stop? From the viewpoint of tau functions, the most natural and general model is the "Orlov-Scherbin" tau function, corresponding to r-constellations with two infinite sets of controlled degrees, and arbitrary color weights. For $r=2$, it is essentially the partition function of the Ising model famously computed by B. Eynard. In recent work with V. Bonzom, S. Charbonnier, E. Garcia-Failde we show that, maybe surprisingly, the topological recursion (TR) holds in full generality for this model. Our proof is a bizarre assembly of nice existing techniques: Albenque-Bouttier slice bijections in genus 0, KP-type techniques to establish TR without internal faces (done in previous work with A. Alexandrov, B. Eynard, J. Harnad) and Eynard-Orantin "deformation techniques" to stuff internal faces back in the game. Combinatorially, I am still puzzled by the fact that this "obviously non quadratic" model satisfies the (quadratric) topological recursion, and maybe this talk will mostly be about trying to share this excitement.

Scaling limit of high-dimensional uniform spanning trees

• Eleanor Archer

Room: Salle 435 A spanning tree of a finite connected graph G is a connected subgraph of G that touches every vertex and contains no cycles. In this talk we will consider uniformly drawn spanning trees of ``high-dimensional'' graphs, and show that, under appropriate rescaling, they converge in distribution as metric-measure spaces to the Brownian CRT. This extends an earlier result of Peres and Revelle (2004) who previously showed a form of finite-dimensional convergence. Based on joint works with Asaf Nachmias and Matan Shalev.

Scaling limit of random cubic planar graphs

• Thomas Lehéricy

Room: Salle 435 In recent years, a lot of progress has been achieved in the understanding of the scaling limit of random planar maps. However, much less is known about the scaling limit of random planar graphs. We show that the scaling limit of random connected cubic planar graphs (respectively multigraphs) is the Brownian sphere. Our approach consists in approximating the metric structure of a random cubic planar graph by that of successive models with modified distances. The bridge between graphs and maps is Whitney’s theorem, which ensures that a 3-connected cubic planar graph is the dual of a simple triangulation. We then extend a framework of Curien and Le Gall to study the modified distances in simple triangulations. Based on a joint work with Marie Albenque and Éric Fusy.