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Enumeration of general planar hypermaps with an alternating boundary

Valentin Baillard, Ariane Carrance, Bertrand Eynard

Abstract

In this paper, we extend the enumerative study of planar hypermaps with an alternating boundary introduced in an earlier work of Bouttier and the second author. In that article, an explicit rational parametrization was obtained for the associated generating function in the case of m-constellations, using a variant of the kernel method. We develop here a new strategy to obtain an algebraic equation in the general case, which includes maps decorated by the Ising model, through a classical many-to-one correspondence. One of the main steps of our strategy is the simultaneous elimination of two catalytic variables. We then apply this strategy to the case of Ising quadrangulations, where we obtain an explicit rational parametrization. As a consequence, we show that some notable properties of the constellations case are no longer satisfied in general.

Enumeration of general planar hypermaps with an alternating boundary

Abstract

In this paper, we extend the enumerative study of planar hypermaps with an alternating boundary introduced in an earlier work of Bouttier and the second author. In that article, an explicit rational parametrization was obtained for the associated generating function in the case of m-constellations, using a variant of the kernel method. We develop here a new strategy to obtain an algebraic equation in the general case, which includes maps decorated by the Ising model, through a classical many-to-one correspondence. One of the main steps of our strategy is the simultaneous elimination of two catalytic variables. We then apply this strategy to the case of Ising quadrangulations, where we obtain an explicit rational parametrization. As a consequence, we show that some notable properties of the constellations case are no longer satisfied in general.

Paper Structure

This paper contains 20 sections, 16 theorems, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Main results
  4. Reminders on related works
  5. Hypermaps with a monochromatic boundary
  6. Previous results on the alternating boundary condition
  7. Preliminaries
  8. General definitions
  9. Correspondence with the Ising model
  10. Tutte decompositions and splitting procedure
  11. Master equation for the alternating boundary condition
  12. Splitting equations
  13. Algebraicity of $\widehat{f}$
  14. Re-derivation of the Tutte equation and kernel system
  15. Explicit solution for Ising quadrangulations
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any $d, \widetilde{d}\geq2$, $\widehat{f}$ is algebraic over $\overline{\mathbb{Q}}((t))(c,t_1,\dots,t_d,\widetilde{t}_1,\dots,\widetilde{t}_{\widetilde{d}},\omega)$. Moreover, we have an explicit strategy to obtain its annihilating polynomial, that does not rely on the kernel method or its gene

Figures (4)

  • Figure 1: Left, a hypermap that admits the boundary conditions $\circ\bullet\circ\bullet\bullet\bullet\circ\bullet\bullet\bullet\circ\circ\circ$ and $\circ\bullet\circ\bullet\bullet\circ\circ\bullet\bullet\bullet\circ\bullet\circ$. Right, a hypermap with a monochromatic boundary.
  • Figure 2: Left, a hypermap with an alternating boundary. Right, a hypermap whose boundary is not alternating.
  • Figure 3: Illustration of the many-to-one correspondence, from hypermaps to Ising-decorated maps, through the contraction of digons.
  • Figure 4: A sketch of the splitting procedure: top left, the boundary edge of type $A$ that we peel. Top right, the case where it is adjacent to a black inner face. Bottom row, the case where it is identified with a boundary edge of type $B$.

Theorems & Definitions (34)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: Theorem Eyn_2003 and 8.3.1 in eynard
  • Theorem 2.2: Explicit parametrization for $m$-constellations, Theorem 1 in bouttier-carrance
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Remark 3.1
  • ...and 24 more