Geodesics in the Brownian map: Strong confluence and geometric structure

• Wei Qian

Room: Salle 435 (UMPA) I will talk about a joint work with Jason Miller (https://arxiv.org/abs/2008.02242) where we establish results on all geodesics in the Brownian map, including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most 5, and the maximal number of geodesics which connect any pair of points is 9. For each k=1,...,9, we obtain the Hausdorff dimension of the pairs of points connected by exactly k geodesics. For k=7,8,9, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame, the union of all of the geodesics in the Brownian map minus their endpoints, has dimension one, the dimension of a single geodesic.

Markovian triangulations and robust convergence to the UIPT

• Thomas Budzinski

Room: Salle 435 (UMPA) We say that a random infinite planar triangulation T is Markovian if for any small triangulation t with boundaries, the probability to observe t around the root of T only depends on the boundaries and the total size of t. Such a property can be expected from the local limits of many natural models of random maps such as the UIPT. We will classify completely infinite Markovian planar triangulations, without any assumption on the number of ends. In particular, there is (almost) no model of multi-ended Markovian triangulation. As an application, we will see the convergence of uniform triangulations to the UIPT is robust under many kinds of perturbations, even when we are not able to count precisely the perturbed models.

A transfer theorem for ∆-analytic functions in several variables and application to multivariate local limit theorems

• Linxiao Chen

Room: Salle 435 (UMPA) In this talk I will present a recipe for reading the asymptotics of a multi-dimensional infinite array of numbers from its generating function. In the bivariate case, this means reading the asymptotics of a_{m,n} as m,n → ∞ and m/n^θ → s (where θ>0 is fixed and s>0 is a variable) from the function A(x,y)= Σ_{m,n≥0} a_{m,n} x^m y^n. The recipe works for functions A which are analytic in a product of ∆-domains, generalizing Flajolet and Odlyzko's classical transfer theorem for univariate functions. I will give some examples of such multivariate ∆-analytic functions which arise naturally from random map models (in particular Ising-decorated triangulations), and show how the new recipe can be used to establish uniform local limit theorems with exotic limit distributions. Previously, similar recipes were available only in the case where θ=1 and the function A is rational. In contrast, our method works for any θ>0 and algebraic function A (under the additional ∆-analyticity condition). I will explain why the old and new methods treat disjoint cases, and how they might be combined together in the future.

Curves, surfaces and intersection

• Hugo Parlier

Room: Salle 435 (UMPA) On closed surfaces of positive genus, through classical work of Dehn, isotopy classes of simple closed curves can be described using their intersection numbers with other curves. Now what if you want to describe all curves in a similar way? This talk will be on joint work with Binbin Xu about this question, and where we end up constructing and studying so-called k-equivalent curves. These are distinct curves that intersect all curves with k self-intersections the same number of times.