A bijection for rooted maps on general surfaces (extended abstract)
Abstrak
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, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable maps of Euler characteristic $2-2h$ and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor $n$<sup>¼</sup> for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size $n$ on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.
Topik & Kata Kunci
Penulis (2)
Guillaume Chapuy
Maciej Dołęga
Akses Cepat
- Tahun Terbit
- 2015
- Sumber Database
- DOAJ
- DOI
- 10.46298/dmtcs.2536
- Akses
- Open Access ✓