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Multitudinous combinatorial structures are counted by generating functionssatisfying a composition scheme $F(z)=G(H(z))$. The corresponding asymptoticanalysis becomes challenging when this scheme is critical (i.e., $G$ and $H$are simultaneously singular). The singular exponents appearing in the Puiseuxexpansions of $G$ and $H$ then dictate the asymptotics.  In this work, we first complement results of Flajolet et al. for a fullfamily of singular exponents of $G$ and $H$. Motivated by many examples (randommappings, planar maps, directed lattice paths), we consider a natural extensionof this scheme, namely $F(z,u)=G(u H(z))M(z)$. We also consider a variant ofthis scheme, which allows us to analyse the number of $H$-components of a givensize in $F$.  These two models lead to a rich world of limit laws, where we identify thekey r\^ole played by a new universal three-parameter law: thebeta-Mittag-Leffler distribution, which is essentially the product of a betaand a Mittag-Leffler distribution. We prove (double) phase transitions,additionally involving Boltzmann and mixed Poisson distributions, with aunified explanation of the associated thresholds. We also obtain momentconvergence and local limit theorems. We end with extensions of the criticalcomposition scheme to a cycle scheme and to the multivariate case, leading toproduct distributions. Applications are presented for random walks, trees(supertrees of trees, increasingly labelled trees, preferential attachmenttrees), triangular P\'olya urns, and the Chinese restaurant process.

Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions