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Asymptotic Transfer in Critical Recursive Composition Schemes

Michael Drmota, Zéphyr Salvy

Abstract

The composition $\mathcal{F} \circ \mathcal{G}$ of two combinatorial classes $\mathcal{F}$ and $\mathcal{G}$ is a standard combinatorial construction and translates into the composition $F(G(z))$ of their corresponding counting generating functions. Such a composition is called critical if $G(ρ_G) = ρ_F$, where $ρ_F$ and $ρ_G$ denote the corresponding radii of convergences of $F$ and $G$, respectively. In this case, both the singular behaviours of $F$ and $G$ influence that of $F\circ G$. Such critical decomposition schemes appear quite frequently in the context of map enumeration. For example by using the block-decomposition one has $M(z) = B(z(1+M(z))^2)$ and $ρ_B = ρ_M (1+M(ρ_M))^2$, where $M(z)$ denotes the generating series of all rooted planar maps and $B(y)$ the generating series of $2$-connected rooted planar maps. This can be extended to multivariate generating functions by taking several statistics into account, for example face counts. Since critical composition schemes show (usually) a condensation phenomenon -- in the above situation this means that there is giant $2$-connected block of linear size and linearly many small blocks -- it is very plausible that statistical properties on $2$-connected maps transfer to corresponding properties of all maps and back. The purpose of the present paper is to make this precise on the level of the singular structure of the corresponding multivariate generating functions. In particular we show that moving $3/2$-singularities transfer. Since such kind of singularities are closely related to central limit theorems of the corresponding statistics this methods provides also a kind of transfer of central limit theorems. Actually this method is quite flexible and is applied to a variety of face and pattern counting statistics in map enumeration.

Asymptotic Transfer in Critical Recursive Composition Schemes

Abstract

The composition of two combinatorial classes and is a standard combinatorial construction and translates into the composition of their corresponding counting generating functions. Such a composition is called critical if , where and denote the corresponding radii of convergences of and , respectively. In this case, both the singular behaviours of and influence that of . Such critical decomposition schemes appear quite frequently in the context of map enumeration. For example by using the block-decomposition one has and , where denotes the generating series of all rooted planar maps and the generating series of -connected rooted planar maps. This can be extended to multivariate generating functions by taking several statistics into account, for example face counts. Since critical composition schemes show (usually) a condensation phenomenon -- in the above situation this means that there is giant -connected block of linear size and linearly many small blocks -- it is very plausible that statistical properties on -connected maps transfer to corresponding properties of all maps and back. The purpose of the present paper is to make this precise on the level of the singular structure of the corresponding multivariate generating functions. In particular we show that moving -singularities transfer. Since such kind of singularities are closely related to central limit theorems of the corresponding statistics this methods provides also a kind of transfer of central limit theorems. Actually this method is quite flexible and is applied to a variety of face and pattern counting statistics in map enumeration.
Paper Structure (18 sections, 13 theorems, 44 equations, 1 figure, 2 tables)

This paper contains 18 sections, 13 theorems, 44 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Plan of the paper
  4. Preliminaries
  5. Singularity notions and a related central limit theorem
  6. Maps and families of maps
  7. Face and Pattern counting
  8. Combinatorics for composition schemes
  9. 2-connected maps
  10. Loopless or bridgeless maps
  11. (2-connected) simple maps
  12. Bipartite maps
  13. Face counts
  14. Transferring singularity analysis
  15. Applications
  16. ...and 3 more sections

Key Result

Proposition 1

Let $f(z,x)$ be the generating function of a combinatorial class $\mathcal{F}$ with $z$ marking the size of the object and $x$ the number of occurrences of a parameter. Then, if $f(z,x)$ has a moving $3/2$-singularity of the form eq32sing2 with $\rho'(1) \ne 0$, and if, in a neighbourhood of $x=1$, where $\mathbb{E}\, X_n \sim \mu n$ and $\mathbb{V}{\rm ar}\, X_n \sim \sigma^2 n$ with $\mu = - {

Figures (1)

  • Figure 1: Four necessarily non-self-intersecting patterns.

Theorems & Definitions (16)

  • Proposition 1
  • Lemma 2
  • Proposition 3
  • Theorem 4
  • Proposition 5
  • Proposition 6
  • Remark 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • ...and 6 more