Asymptotic Transfer in Critical Recursive Composition Schemes

Abstract

The composition ℱ∘𝒢 of two combinatorial classes ℱ and 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 convergence of F and G, respectively. In this case, both the singular behaviours of F and G influence that of F∘G. Such critical composition schemes arise frequently in 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 function of all rooted planar maps and B(y) the generating functions 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 exhibit (usually) a condensation phenomenon - in the above situation this means that there is a 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 at the level of the singular structure of the corresponding multivariate generating functions. In particular, we show that moving 3/2-singularities transfer. Since such singularities are closely related to central limit theorems of the corresponding statistics, this method also provides 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.