Related papers: Quadrangulations with no pendant vertices

Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under…

By generalizing the path method, we show that nonlinear spectral gaps of a finite connected graph are uniformly bounded from below by a positive constant which is independent of the target metric space. We apply our result to an $r$-ball…

We study the behaviour of random labelled and unlabelled cographs with n vertices as n tends to infinity. Our main result is a novel probabilistic limit in the space of graphons.

Fix $c\in (0,1)$ and let $\Gamma$ be a $\lfloor c n\rfloor$-regular digraph on $n$ vertices drawn uniformly at random. We prove that when $n$ is large, the (non-symmetric) adjacency matrix $M$ of $\Gamma$ is invertible with high…

We introduce and study the uniform infinite planar quadrangulation (UIPQ) with a boundary via an extension of the construction of arXiv:1201.1052. We then relate this object to its simple boundary analog using a pruning procedure. This…

This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to…

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n \ge 1$, a random quadrangulation $\q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations…

We prove local convergence of the $t$-PNG model with zero boundary to the stationary $t$-PNG model, confirming a recent conjecture of Drillick and Lin (2024). The stationary $t$-PNG model is the one with both left and bottom boundaries of…

We show that the planar normalized orthogonal polynomials $P_{m,n}(z)$ of degree $n$ with respect to an exponentially varying planar measure $\mathrm{e}^{-2mQ}\mathrm{dA}$ enjoy an asymptotic expansion \[ P_{m,n}(z)\sim…

This article presents new enumerative results related to symmetric planar maps. In the first part a new way of enumerating rooted simple quadrangulations and rooted simple triangulations is presented, based on the description of two…

We give a new proof that the Poisson boundary of a planar graph coincides with the boundary of its square tiling and with the boundary of its circle packing, originally proven by Georgakopoulos and Angel, Barlow, Gurel-Gurevich and Nachmias…

We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of…

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than 180 degrees. In this paper we prove that the opposite statement is also true, namely that planar…

This article surveys results for Riemannian manifolds of positive and non-negative sectional curvature with symmetries.

We prove that the total curvature of a planar graph with nonnegative combinatorial curvature is at least $\frac{1}{12}$ if it is positive. Moreover, we classify the metric structures of ambient polygonal surfaces for planar graphs attaining…

We study a large class of Bernoulli percolation models on random lattices of the half- plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact value of the site percolation threshold…

We show that for a metric space with an even number of points there is a 1-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the…

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This…

We introduce and study a universal model of random geometry in two dimensions. To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class of graphs with a given size $n$, for…

We give a complete classification of homogeneous $(\alpha,\beta)$-metrics with positive flag curvature and vanishing S-curvature.