Blossoming bijection for bipartite maps: a new approach via orientations and applications to the Ising model

TL;DR

This work develops a new bijective framework for enumerating bipartite planar maps with controlled black/white vertex degrees by leveraging alpha-d orientations and blossoming-tree techniques. It unifies and extends the Bousquet-Mélou–Schaeffer bijection within the Albenque–Poulalhon scheme, introducing alpha-d orientations, alpha-d-trees, and their closures, and generalizes to d-trumpets and d-cornets with complete closures linking charged trees to annular map structures. The authors derive a rational and Lagrangian parametrization for quartic maps decorated with an Ising model, establishing nonnegative-integer coefficients that enable probabilistic analyses and explicit enumeration. They also provide explicit quartic-case computations, clarifying connections to BDG and other bijections, and offering new tools for studying Ising-type models on maps across genera.

Abstract

We develop a new bijective framework for the enumeration of bipartite planar maps with control on the degree distribution of black and white vertices. Our approach builds on the blossoming-tree paradigm, introducing a family of orientations on bipartite maps that extends Eulerian and quasi-Eulerian orientations and connects the bijection of Bousquet-Mélou and Schaeffer to the general scheme of Albenque and Poulalhon. This enables us to generalize the Bousquet-Mélou and Schaeffer's bijection to several families of bipartite maps. As an application, we also derive a rational and Lagrangian parametrization with positive integer coefficients for the generating series of quartic maps equipped with an Ising model, which is key to the probabilistic study of these maps.

Figures (20)

• Figure 1: A blossoming bipartite tree with charge 4 (left), its complete closure with a additional black vertex of degree 4 (middle), and the corresponding dual hypermap with a additional marked face of degree 4 (right).
• Figure 2: Three representations of the same plane map: as a map embedded in the sphere with a marked dashed face (left), and as a map embedded in the plane with the rooted corner in the outer face (middle), or with the marked face as the outer face (right). In the rest of this article, we will always use the last representation.
• Figure 3: A bipartite map with a proper 2-coloring of its vertices (left), its dual with the correponding coloring of its faces (middle), and with the canonical direction of its edges and its directed geodesic labeling (right). The bipartite map verifies $\Delta_\circ=4$ and $\Delta_\bullet=5$.
• Figure 4: A map with a non-minimal (see the orange-highlighted cycle) and non-accessible (see the pink-highlighted vertex) 3-fractional orientation (left), and a bipartite map with its minimal $\alpha_4$-orientation (right).
• Figure 5: A blossoming tree of charge $2$ with a $3$-fractional orientation (left). The same tree with the matching of stems in dashed lines (middle). Its closure, which is a blossoming map with two unmatched closing stems (right).

Theorems & Definitions (69)

• Proposition 1.1
• Theorem 1.2: AlbenquePoulalhon_Generic, Corollary 2.4
• Remark 2.2
• proof
• proof
• Theorem 2.5
• Remark 2.6
• Definition 2.7
• Lemma 2.8
• proof