We consider the (non-spatial) coalescent model (sometimes called the Marcus-Lushnikov model), starting with N particles with mass one each, where each two particles coalesce after independent exponentially distributed times. The corresponding coagulation kernel is multiplicative in the two masses, hence the coalescent is also called multiplicative. There are strong relations with the time-dependent Erdös-Renyi graph. We work in the thermodynamic limit N→∞ at a fixed time t and derive a joint large-deviations principle for all relevant quantities (microscopic, mesoscopic and macroscopic particle sizes) with an explicit rate function. We deduce laws of large numbers and in particular derive from that the well-known phase transition at time t=1, the time at which a macroscopic particle appears, as well as the well-known Smoluchowski characterisation of the statistics of the finite-sized particles. Joint work with Luisa Andreis and Robert Patterson.
Wolfgang König received his Ph.D. from University of Zurich in 1994 and his Habilitation in 2000 from TU Berlin. He was Professor at Leipzig University from 2004 till 2009 and was then appointed head of a research group at WIAS and Professor for Probability at TU Berlin. Wolfgang König is an expert on the intermittency analysis of the heat equation with random potential (also called parabolic Anderson model) and various models from probabilistic statistical mechanics, like intersections and non-intersections of Brownian motions, many-body systems with and without kinetic energy, and more. Recently he extended his interests towards stochastic geometry with applications in telecommunications.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai