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Journée organisée par Cyril Banderier et Philippe Marchal dans le cadre de la structure fédérative mathSTIC de l'université de Paris-Nord.
Le programme officiel est archivé sur https://lipn.fr/~cb/Seminaires/.
The study of random combinatorial structures and their limits is a growing field at the interface of combinatorics, probability theory, and mathematical physics. Planar graphs are a prominent example of such structures, yet important problems concerning their asymptotic shape remain open. This talk highlights open conjectures and reviews recent results, in particular the discovery of a Uniform Infinite Planar Graph (UIPG) as their quenched local limit.
We consider simply generated random plane trees with n vertices and kn leaves, sampled from a sequence of weights. Motivated by questions on random planar maps, we will focus on the asymptotic behaviour of the largest degree. Precisely we will give conditions on both the number of leaves and the weight sequence that ensure the convergence in distribution of the associated Łukasiewicz path (or depth-first walk) to the Brownian excursion. This should also provide a first step towards the convergence of the height or contour function of the trees. The proof scheme is to reduce step by step to simpler and simpler objects and we will discuss excursion and bridge paths, non decreasing paths conditioned by their tip, and finally estimates of the form of the local limit theorem which may be of independent interest. Based on a joint work with Igor Kortchemski.
We will first introduce translation surfaces, which are Riemann surfaces built from gluing polygons in the plane via translations. The torus is the most basic example of a translation surface. The closed geodesics we count are called saddle connections, and are found by following geodesics which start and end at a marked point. In the case of the torus, the saddle connections correspond to pairs of integers (a,b) which are coprime to each other. We will present probabilistic results counting saddle connections with length conditions, as well as counting pairs of saddle connections with various pairing conditions. We will finish with highlighting the open questions and difficulties of counting triples of closed geodesics.
The study of maps (graphs embedded into surfaces) is a rich subject at the crossroads of mathematics, computer science and theoretical physics. In this talk I will review the slice decomposition of planar maps : it started as a reformulation of some (by now) classical bijections between planar maps and trees, but then evolved into a general framework applicable to many families of maps and to their scaling limits. For simplicity I will restrict to the easiest setting of bipartite maps with controlled face degrees. I will explain how slice decomposition allows to enumerate pointed rooted maps, then "cylinders" and "pairs of pants" (planar maps with two and three boundaries, respectively). The talk is based on joint works with Emmanuel Guitter and Grégory Miermont.
Catalytic equations appear in several combinatorial applications, most notably in the numeration of lattice path and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of maps of size n in various planar map classes grows asymptotically like c.n^{−5/2} γ^n, for suitable positive constants c and γ. Essentially we have to distinguish between linear catalytic equations (where the subexponential growth is n^{−3/2}) and non-linear catalytic equations (where we have n^{−5/2} as in planar maps). Furthermore we provide a quite general central limit theorem for parameters that can be encoded by catalytic functional equations, even when they are not positive. Joint work with Marc Noy and Guan-Ru Yu.