Examination procedure
Assessment will take place through a written test and an oral exam.
Examination procedure notes
Alternative or supplementary assessment methods (e.g., a seminar presentation) may be agreed upon, particularly for PhD students.
Prerequisites
The course is recommended for both master's students and PhD students, and potentially for interested third-year undergraduate students. There are no significant technical prerequisites other than the course Elements of Probability and Statistics (052AA); however, the course Probability (070AA) is strongly recommended. This course is intended in part as a complement to Probability on Graphs, but the two courses may be taken in either order.
Syllabus
The course will cover several aspects of the theory of random graphs. First, we will discuss in some detail the model of Bienaimé–Galton–Watson trees, together with their local and scaling limits. We shall then turn to the Erdős–Rényi model and study its phase transition(s), as well as some related examples and generalisations such as the configuration model, inhomogeneous random graphs, etc. Finally, we will delve into the theory of random planar maps and their limits, highlighting combinatorial techniques for enumeration and ties to different areas of Mathematics and Physics.
Bibliographical references
N. Curien, A random walk among random graphs, Cours Spécialisés, 31, Soc. Math. France, Paris, 2025; MR5008856
R. van der Hofstad, Random graphs and complex networks., Cambridge Series in Statistical and Probabilistic Mathematics, [43], Cambridge Univ. Press, Cambridge, 2017; MR3617364
J.-F. Le Gall and G. Miermont, Scaling limits of random trees and planar maps, in Probability and statistical physics in two and more dimensions, 155--211, Clay Math. Proc., 15, Amer. Math. Soc.; MR3025391