Explore a comprehensive lecture on phase transitions of composition schemes and their universal limit laws presented by Cyril Banderier from CNRS, Université Paris Nord. Delve into the analysis of combinatorial structures counted by generating functions satisfying composition schemes, focusing on critical cases where asymptotic analysis becomes challenging. Examine natural extensions of these schemes, including F(z,u)=G(uH(z))M(z) and variants analyzing H-component sizes in F. Discover a rich world of limit laws involving Mittag-Leffler distributions, stable distributions, and phase transitions with Boltzmann and mixed Poisson distributions. Learn about the universality of phase transitions with n^(1/3) window sizes. Apply these concepts to random walks, trees (supertrees, increasingly labelled trees, preferential attachment trees), and extensions of works by Flajolet, Pitman, and Janson. Gain insights from this joint work with Markus Kuba and Michael Wallner, presented at the Institut des Hautes Etudes Scientifiques (IHES) in a 47-minute talk. Read more