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

TL;DR

This paper introduces a new bijective framework for enumerating bipartite planar maps using orientations, extending existing bijections, and applies it to derive a parametrization for maps with an Ising model.

Contribution

It develops a novel orientation-based bijection framework for bipartite maps, generalizing previous bijections and enabling new enumerative and probabilistic analyses.

Findings

Extended bijection to various bipartite map families

Derived a rational parametrization for quartic maps with Ising model

Facilitated probabilistic study of these maps

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-M\'elou and Schaeffer to the general scheme of Albenque and Poulalhon. This enables us to generalize the Bousquet-M\'elou 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.