Related papers: Asymptotics for rooted planar maps and scaling lim…

We study the asymptotic distribution, as the volume parameter goes to 1, of the peak (largest part) of finite- or slowly-growing-width cylindric plane partitions weighted by their trace, seam, and volume. There are two natural asymptotic…

The method of asymptotics beyond all orders (ABAO) is known to be a useful tool to investigate separatrix splitting of several maps. For a class of simplectic maps, the form of maps is shown to be restricted by the conditions for the ABAO…

We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the "peano curve" surrounding the tree of…

We investigate conditions for the existence of the limiting conditional distribution of a bivariate random vector when one component becomes large. We revisit the existing literature on the topic, and present some new sufficient conditions.…

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations.…

An $(a,b)$-biregular bipartite graph is a bipartite graph with bipartition $(X, Y)$ such that each vertex in $X$ has degree $a$ and each vertex in $Y$ has degree $b$. By the bipartite expander mixing lemma, biregular bipartite graphs have…

We study the scaling limits of looptrees associated with Bienaym\'e--Galton--Watson (BGW) trees, that are obtained by replacing every vertex of the tree by a "cycle" whose size is its degree. First, we consider BGW trees whose offspring…

We present a bijection for toroidal maps that are essentially $3$-connected ($3$-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite toroidal maps with all faces of…

We start by studying a peeling process on finite random planar maps with faces of arbitrary degrees determined by a general weight sequence, which satisfies an admissibility criterion. The corresponding perimeter process is identified as a…

We consider the problem of computing the distance-dependent two-point function of general planar maps and hypermaps, i.e. the problem of counting such maps with two marked points at a prescribed distance. The maps considered here may have…

We present a technique to study normalizing strategies when termination is asymptotic, that is, it appears as a limit, as opposite to reaching a normal form in a finite number of steps. Asymptotic termination occurs in several settings,…

It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is…

We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of relatively hyperbolic groups (resp. asymptotic cones of groups containing a cut-point) only depend on the bilipschitz equivalence types of the…

We introduce a notion of bipartite minors and prove a bipartite analog of Wagner's theorem: a bipartite graph is planar if and only if it does not contain $K_{3,3}$ as a bipartite minor. Similarly, we provide a forbidden minor…

We determine the colored patterns that appear in any $2$-edge coloring of $K_{n,n}$, with $n$ large enough and with sufficient edges in each color. We prove the existence of a positive integer $z_2$ such that any $2$-edge coloring of…

We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the $N\times N$ Hermitian random matrix model with cubic potential. We prove that the recurrence…

We show that large critical multi-type Galton-Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analoguous to Kesten's infinite monotype Galton-Watson tree. This is proven when we…

Let $e^{2\pi i\Q}$ denote the set of roots of unity. We consider subsets $E\subset e^{2\pi i\Q}$ that are quasi-independent or algebraically independent (as subsets of the discrete plane). A bijective map on $e^{2\pi i\Q}$ preserves the…

We study asymptotics of the spectral measure of a randomly chosen partial automorphism of a rooted tree. To every partial automorphism $x$ we assign its action matrix $A_x$. It is shown that the uniform distribution on eigenvalues of $A_x$…

We define the (random) $k$-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon except now a node must be cut $k$ times before it is destroyed. The…