Enumerating planar stuffed maps as hypertrees of mobiles

TL;DR

The work addresses enumerating planar stuffed maps by establishing a bijection with hypertrees called hypermobiles, extending the BDFG correspondence to multi-boundary components. It develops a generating-function framework in which stuffed maps correspond to algebraic and functional equations, including a tree equation governing a key parameter $\gamma$ and a gasket-based functional relation linking stuffed maps to ordinary maps. The main result is a bijection between rooted pointed branch-pointed stuffed maps and hypermobiles, enabling exact enumeration and algebraicity results; an explicit example for bridged quadrangulations demonstrates the method. The study lays groundwork for understanding the geometry and scaling of stuffed maps with potential applications to random surfaces and matrix/tensor models.

Abstract

A planar stuffed map is an embedding of a graph into the 2-sphere $S^{2}$, considered up to orientation-preserving homeomorphisms, such that the complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to $S^{2}$ with multiple boundaries. This is a generalization of planar maps whose complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to a disc. The main goal of this work is to construct a bijection between bipartite planar stuffed maps and collections of integer-labelled trees connected by hyperedges such that they form a hypertree, called hypermobiles. This bijection directly generalizes the Bouttier-Di Franceso-Guitter bijection between bipartite planar maps and mobiles. Additionally, we show that the generating functions of these trees of mobiles satisfy both an algebraic equation, generalizing the case of ordinary planar maps, and a new functional equation. As an example, we explicitly enumerate a class of stuffed quadrangulations.

Figures (5)

• Figure 1: Sub-figure (a) is an example of an bipartite planar rooted (ordinary) map. Sub-figure (b) is an example of a (bipartite planar) rooted stuffed map. The grey shaded region distinguishes a 2-cell with more than one boundary. In particular, this map is constructed from two such 2-cells: one with boundaries of lengths eight, six, and four, and the other with two boundaries both of length four.
• Figure 2: The associated hypertree to the stuffed map in Figure 1 (b). The middle vertex corresponds to the gasket.
• Figure 3: Sub-figure (a) is an example of an mobile. Sub-figure (b) is an example of a hypermobile, where each hyperedge is represented by a different colour.
• Figure 4: Stages of the construction of a hypermobile from the BPS map seen in Figure 1 (b). Sub-figure (a) is a BPS map whose vertices are labelled with the distance of the source and whose edges next to gate vertices are connected by hyperedges. In sub-figure (b), edges are drawn according to the same rules as the BDFG bijection on each component. Sub-figure (c) is the resulting hypermobile after removing all the original edges and branches, and shifting labels.
• Figure 5: Stages of the construction of a BPS map from the hypermobile seen in Figure 2 (b). Sub-figure (a) and (b) have additional edges and vertices drawn according to the BDFG rules on each mobile. Sub-figure (c) is the resulting BPS map.

Theorems & Definitions (12)

• Definition 2.1
• Definition 2.2
• Remark
• Theorem 2.1
• proof
• Proposition 1
• proof
• Remark
• Proposition 2: Tree equation
• proof