Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions

Multitudinous combinatorial structures are counted by generating functions satisfying a composition scheme $F(z)=G(H(z))$. The corresponding asymptotic analysis becomes challenging when this scheme is critical (i.e., $G$ and $H$ are simultaneously singular). The singular exponents appearing in the Puiseux expansions of $G$ and $H$ then dictate the asymptotics. In this work, we first complement results of Flajolet et al. for a full family of singular exponents of $G$ and $H$. We identify the arising limit laws (for the number of $H$-components in $F$) and prove moment convergence. Then, motivated by many examples (random mappings, planar maps, directed lattice paths), we consider a natural extension of this scheme, namely $F(z)=G(H(z))M(z)$. We discuss the number of $H$-components of a given size in $F$; this leads to a nice world of limit laws involving products of beta distributions and of Mittag-Leffler distributions. We also obtain continuous to discrete phase transitions involving mixed Poisson distributions, giving an unified explanation of the associated thresholds. We end with extensions of the critical composition scheme to a cycle scheme and to the multivariate case, leading again to product distributions. Applications are presented for random walks, trees (supertrees of trees, increasingly labelled trees, preferential attachment trees), triangular P\'olya urns, and the Chinese restaurant process.