I have defended my habilitation thesis on September 21st. You can find here the manuscript, the abstract and the composition of the jury.
Random combinatorial structures form an active field of research at the interface between combinatorics and probability theory. From a theoretical point of view, some of the main objectives are to develop general methods to find limiting distributions of functionals on random objects, and to understand universality classes of their scaling limits. This habilitation thesis reports on three series of papers fitting in this general effort. In the first part, we study models of random Young diagrams arising from representation and symmetric function theories. In the second part, we develop the theory of dependency graphs, used to prove asymptotic normality of some functionals, mainly of substructure counts in random combinatorial objects. In the last part, we present a new universal scaling limit for pattern-avoiding permutations, the Brownian separable permuton.