A bijection for rooted maps on general surfaces

TL;DR

This paper generalizes a bijection between rooted maps and labeled maps from orientable surfaces to all surfaces, providing new combinatorial insights, universal scaling limits, and extending bijections to study Brownian surfaces.

Contribution

It extends the Marcus-Schaeffer bijection to all surfaces, offering a unified combinatorial framework for orientable and non-orientable maps and their generating functions.

Findings

Uniform combinatorial interpretation of counting exponents

Algebraicity of generating functions for all surfaces

Universal scaling limits for distances in maps

Abstract

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 general construction requires new ideas and is more delicate than the special orientable case, but it carries the same information. In particular, it leads to a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rooted connected maps of Euler characteristic $2-2h$, and of the algebraicity of their generating functions, similar to the one previously obtained in the orientable case via the Marcus-Schaeffer bijection. It also shows that the renormalization factor $n^{1/4}$ for distances between vertices is universal for maps on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation on any fixed surface converge in distribution when the size $n$ tends to infinity. Finally, we extend the Miermont and Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our construction opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.