Planar maps as labeled mobiles

Blocked edges on Eulerian maps and mobiles: Application to spanning trees, hard particles and the Ising model

We introduce Eulerian maps with blocked edges as a general way to implement statistical matter models on random maps by a modification of intrinsic distances. We show how to code these dressed maps by means of mobiles, i.e. decorated trees with labeled vertices, leading to a closed system of recursion relations for their generating functions. We discuss particular solvable cases in detail, as well as various applications of our method to several statistical systems such as spanning trees on quadrangulations, mutually excluding particles on Eulerian triangulations or the Ising model on quadrang…

Vertex Degrees in Planar Maps

We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss some possible extension to maps of higher genus.

Limit Laws of Planar Maps with Prescribed Vertex Degrees

We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers $D$. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.

Local limit of labeled trees and expected volume growth in a random quadrangulation

Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton--Watson trees. As a consequence, we find that the expected volume of the ball of radius $r$ around a marked point in the limit random surface is $Θ(r^4)$.

带标号树与Dyck铺砌的计数

Enumeration of labeled trees and Dyck tilings

本文利用Dyck铺砌研究平面带标号有根树的偏序集，证明其生成函数在特定条件下可因式分解，并通过两种分解方式建立有理Dyck铺砌的权重关系与偏序结构。

关于计数平面紧映射的拟多项式

On quasi-polynomials counting planar tight maps

本文利用平面映射的切片分解与双射方法，给出了计数具有指定面度数的平面紧映射的拟多项式显式公式，并将Tutte切片公式推广至所有非二分映射。

Planar maps and random partitions

This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their metric properties: from a general map-mobile bijection, we compute the three-point function of quadrangulations, before discussing the connection with continued fractions. Chapter 2 presents the slice decomposition, a unified bijective approach that applies notably to irreducible maps. Chapter 3 concerns the $O(n)$ loop model on planar maps: by a combinatorial decomposition, we obtain the phase diagr…

Planar trees, free nonassociative algebras, invariants, and elliptic integrals

We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows to apply combinatorial techniques to study their Hilbert series and the asymptotics of their coefficients. These algebras satisfy the Nielsen-Schreier property and their subalgebras are also free. Then, over a field of characteristic 0, we investigate the subalgebras of invariants under the action of a linear group, their sets of free generators and their Hi…

Edit Distance between Unlabeled Ordered Trees

There exists a bijection between one stack sortable permutations --permutations which avoid the pattern 231-- and planar trees. We define an edit distance between permutations which is coherent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for $(231)$ avoiding permutations. Moreover, we obtain the generating function of the edit distance between ordered trees and some special ones. For the general case we show that the mean edit distance between a planar tree and all other planar trees is at least $n/l…

Hopf algebras of parking functions and decorated planar trees

We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We…