The degree distribution in bipartite planar maps: applications to the Ising model

TL;DR

This paper characterizes the generating function of bipartite planar maps based on vertex degree distribution and applies it to solve the Ising and hard particle models on random planar lattices, extending previous matrix integral results.

Contribution

It provides a purely combinatorial approach to analyze bipartite planar maps and explains the algebraic nature of solutions for the Ising and hard particle models.

Findings

Generated functions characterized for bipartite planar maps.

Extended solutions for Ising and hard particle models.

Proved algebraic solutions for maps of bounded degree.

Abstract

We characterize the generating function of bipartite planar maps counted according to the degree distribution of their black and white vertices. This result is applied to the solution of the hard particle and Ising models on random planar lattices. We thus recover and extend some results previously obtained by means of matrix integrals. Proofs are purely combinatorial and rely on the idea that planar maps are conjugacy classes of trees. In particular, these trees explain why the solutions of the Ising and hard particle models on maps of bounded degree are always algebraic.