In this talk, I will present recent results, obtained in collaboration with Laurent Ménard, about the geometry of spin clusters in Ising-decorated triangulations, and build on previously work obtained in collaboration with Laurent Ménard and Gilles Schaeffer.
In this model, triangulations are sampled together with a spin configuration on their vertices, with a probability biased by their number of monochromatic edges, via a parameter nu. The fact that there exists a combinatorial critical value for this model has been initially established in the physics literature by Kazakov and was rederived by combinatorial methods by Bousquet-Mélou and Schaeffer, and Bouttier, Di Francesco and Guitter.
Here, we give geometric evidence of that this model undergoes a phase transition by studying the volume and perimeter of its monochromatic clusters. In particular, we establish that, when nu is critical or subcritical, the cluster of the root is finite almost surely, and is infinite with positive probability for nu supercritical.[-]

Bijections between planar maps and tree-like structures have been proven to play a crucial role in understanding the geometry of large random planar maps. Perhaps the most popular (and useful) bijections fit into two categories: bijections between maps and labeled trees and bijections between maps and blossoming trees. They were popularized in the late nineties in the important contribution of Schaeffer and they have been widely developed since then. It is natural to ask whether these bijections still hold when the underlying surface is no longer the sphere but any two-dimensional compact manifold? In this case trees are replaced by maps on a given surface with only one face and while the construction of Schaefer of the labeled-type bijection works independently on genus (but crucially depending on the assumption of orientability) his construction of the blossoming-type bijection was known only in the planar case. We will discuss a (recent?) development of these bijections that extends them to all compact two-dimensional manifolds. I will quickly review my previous joint work with Chapuy and its extension due to Bettinelli which treats the labeled-type bijection and will focus on a more recent work joint with Lepoutre which extends the blossoming-type bijection to non-oriented surfaces.[-]

In a recent paper, the speaker and M.I. Ostrovskii developed a new metric embedding method based on the theory of equal-signs-additive (ESA) sequences developed by Brunel and Sucheston in 1970's. This method was used to construct bilipschitz embeddings of diamond and Laakso graphs with an arbitrary finite number of branches into any non-superreflexive Banach space with a uniform bound on distortions that is independent of the number of branches.
In this talk we will outline a proof that the above mentioned embeddability results cannot be obtained using the embedding method which was used for trees by Bourgain (1986) and for binary branching diamonds and Laakso graphs by Johnson and Schechtman (2009), and which is based on a classical James' characterization of superreflexivity (the factorization between the summing basis and the unit vector basis of $\ell_1$). Our proof uses a “self-improvement” argument and the Ramsey theorem.
Joint work with M.I. Ostrovskii.[-]

Shortcut graphs are graphs in which long enough cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory including: systolic and quadric groups (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), cocompactly cubulated groups, hyperbolic groups, Coxeter groups and the Baumslag-Solitar group BS(1,2). Most of these examples satisfy a strong form of the shortcut property. I will discuss some of these examples as well as some general constructions and properties of shortcut graphs and groups.[-]