28 research outputs found
A simple explicit bijection between (n,2) Gog and Magog trapezoids
A sub-problem of the open problem of finding an explicit bijection between
alternating sign matrices and totally symmetric self-complementary plane
partitions consists in finding an explicit bijection between so-called
Gog trapezoids and Magog trapezoids. A quite involved bijection was
found by Biane and Cheballah in the case . We give here a simpler
bijection for this case
Compact Brownian surfaces I. Brownian disks
We show that, under certain natural assumptions, large random plane bipartite
maps with a boundary converge after rescaling to a one-parameter family
(, ) of random metric spaces homeomorphic to the
closed unit disk of , the space being called the
Brownian disk of perimeter and unit area. These results can be seen as an
extension of the convergence of uniform plane quadrangulations to the Brownian
map, which intuitively corresponds to the limit case where . Similar
results are obtained for maps following a Boltzmann distribution, in which the
perimeter is fixed but the area is random
Increasing Forests and Quadrangulations via a Bijective Approach
In this work, we expose four bijections each allowing to increase (or
decrease) one parameter in either uniform random forests with a fixed number of
edges and trees, or quadrangulations with a boundary having a fixed number of
faces and a fixed boundary length. In particular, this gives a way to sample a
uniform quadrangulation with n + 1 faces from a uniform quadrangulation with n
faces or a uniform forest with n+1 edges and p trees from a uniform forest with
n edges and p trees
Convergence of uniform noncrossing partitions toward the Brownian triangulation
International audienceWe give a short proof that a uniform noncrossing partition of the regular n-gon weakly converges toward Aldous's Brownian triangulation of the disk, in the sense of the Hausdorff topology. This result was first obtained by Curien & Kortchemski, using a more complicated encoding. Thanks to a result of Marchal on strong convergence of Dyck paths toward the Brownian excursion, we furthermore give an algorithm that allows to recursively construct a sequence of uniform noncrossing partitions for which the previous convergence holds almost surely. In addition, we also treat the case of uniform noncrossing pair partitions of even-sided polygons
A bijection for nonorientable general maps
International audienceWe give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.We give a different presentation of a recent bijection due to Chapuy and Dołęga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier-Di Francesco-Guitter-like generalization of the Cori-Vauquelin-Schaeffer bijection in the context of general nonori-entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao. Résumé. On donne une présentation différente d'une bijection récente due à Chapuy et Dołęga pour les quadrangula-tions biparties non-orientables et on l'étend au cas des cartes générales non-orientables. Cela peut se voir comme une généralisation à la Bouttier-Di Francesco-Guitter de la bijection de Cori-Vauquelin-Schaeffer dans le contexte des surfaces non-orientables générales. Dans le cas particulier des triangulations, les objets codant prennent une forme particulièrement simple et on retrouve la fameuse formule d'énumération asymptotique de Gao
Scaling limit of random planar quadrangulations with a boundary
International audienceWe discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence of integers such that tends to some . For every , we call a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having faces and half-edges on the boundary. For , we view as a metric space by endowing its set of vertices with the graph metric, rescaled by . We show that this metric space converges in distribution, at least along some subsequence, toward a limiting random metric space, in the sense of the Gromov--Hausdorff topology. We show that the limiting metric space is almost surely a space of Hausdorff dimension with a boundary of Hausdorff dimension that is homeomorphic to the two-dimensional disc. For , the same convergence holds without extraction and the limit is the so-called Brownian map. For , the proper scaling becomes and we obtain a convergence toward Aldous's CRT
Random maps
International audienceThis is a quick survey on some recent works done in the field of random maps