The seminar is sponsored by NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.
We consider a discrete model of coagulation, where a large number of particles are initially given a prescribed number of arms, used to create links between particles. Arms are successively chosen uniformly at random and bound two by two, unless they belong to "large" clusters. In that sense, the large clusters are frozen and become inactive. This is a discrete version of a continuous model of coagulation called Smoluchowski's equation with limited aggregations. We study the graph structure induced by this discrete model, and describe typical "small" clusters. We show that there is a fixed time T such that, before time T, a typical cluster is a subcritical Galton-Watson tree, whereas after time T, a typical cluster is a critical Galton-Watson tree. In that sense, we observe a phenomenon called self-organized criticality. This study extensively relies on fine results concerning a class of random graphs in the critical window.
Raoul Normand is a postdoctoral fellow at the Academia Sinica in Taipei. He received his Ph.D. from University Paris 6 in 2011. His research interests include models from statistical physics, such as coagulation processes, random graphs and random matrices, as well as Monte Carlo methods.