Abstract

A planar hypermap with a boundary is defined as a planar map with a boundary, endowed with a proper bicoloring of the inner faces. The boundary is said alternating if the colors of the incident inner faces alternate along its contour. In this paper we consider the problem of counting planar hypermaps with an alternating boundary, according to the perimeter and to the degree distribution of inner faces of each color. The problem is translated into a functional equation with a catalytic variable determining the corresponding generating function.
 In the case of constellations—hypermaps whose all inner faces of a given color have degree $m\geq 2$, and whose all other inner faces have a degree multiple of $m$—we completely solve the functional equation, and show that the generating function is algebraic and admits an explicit rational parametrization.
 We finally specialize to the case of Eulerian triangulations—hypermaps whose all inner faces have degree $3$—and compute asymptotics which are needed in another work by the second author, to prove the convergence of rescaled planar Eulerian triangulations to the Brownian map.

Highlights

  • Our approach relies on the framework of functional equations with catalytic variables arising from the recursive “peeling” decomposition of maps

  • This framework is closely related to the so-called topological recursion for maps: in a nutshell, the core idea is that, given a family of maps defined by some “bulk” conditions, the corresponding generating functions for all sorts of topologies and boundary conditions can be constructed as functions defined on a common algebraic curve known as the “spectral curve”

  • In the context of hypermaps, this idea is discussed in [21, Chapter 8], and the spectral curve is nothing but the algebraic curve (x(z), y(z))z∈C∪{∞} of genus zero that we have been using in this paper

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Summary

Introduction

Given a nonnegative integer r, let Ar denote the generating function of m-constellations with an alternating boundary of length 2r, counted with a weight t per vertex and a weight xi per white inner face of degree mi for every i 1. P nonnegative integers, denote by Bn,p the number of Eulerian triangulations with a semi-simple alternating boundary of length 2p and with n black triangles. By analogy with similar families of random maps, we call Z(p) the partition function of critical Boltzmann Eulerian triangulations with a semi-simple alternating boundary of length 2p. Auxiliary material is left in the appendices: in Appendix A we compute the generating function of rooted m-constellations via two approaches; Appendix B deals with the rational parametrization of the generating function of hypermaps with a monochromatic boundary of prescribed length

Reminders on the enumeration of constellations
A functional equation for hypermaps with an alternating boundary
Simplification in the case of m-constellations
Asymptotics for Eulerian triangulations
Conclusion
A The generating function of rooted m-constellations
B Rational parametrizations for monochromatic boundaries
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