Résumé : We will discuss the emerging substructures in graphs with degree constraints, digraphs, acyclic digraphs, and 2-CNF below the point of their respective phase transitions. While long time ago, probabilistic method allowed to establish the location of the transition points, with some very rough properties, more and more refined results gradually become available. With analytic combinatorics as a main tool, we will see what exactly is happening when the structure approaches its boiling point. Based on joint works with Élie de Panafieu and Vlady Ravelomanana.
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