Asymptotic Transfer in Critical Recursive Composition Schemes

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.