Seminario di Probabilità: “A large-deviations approach to the multiplicative coagulation process”

Venerdì 12 Ottobre 2018, ore 11:30 - Sala Riunioni VII Piano - Luisa Andreis


Venerdì 12 Ottobre 2018 alle ore 11:30 in Sala Riunioni VII Piano, Luisa Andreis (Weierstraß-Institut für Angewandte Analysis und Stochastik - WIAS) terrà un seminario dal titolo “A large-deviations approach to the multiplicative coagulation process”.

At least since the days of Smoluchovski, there is a desire to understand the behaviour of large particle systems that undergo chemical reactions of coagulation type. One of the phenomena that attracts much attention is the question for the existence of a phase transition of gelation type, i.e., the appearance of a particle of macroscopic size in the system. In this talk, we consider the (non-spatial) coagulating model (sometimes called the Marcus-Lushnikov model), starting with N particles with mass one each, where each two particles coagulate after independent exponentially distributed times that depend on a given coagulation kernel, function of the two masses. We focus on the case in which the corresponding coagulation kernel is multiplicative in the two masses, hence the process is identified as the multiplicative coagulation process. This case is of particular interest also for its strong relations with the time dependent Erdös-Rényi random graph. We work for fixed time t > 0 and derive, for the number N of initial particles going to infinity, a joint large-deviations principle for all relevant quantities in the system (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 (the so-called gel) appears, as well as the Smoluchovski characterisation of the statistics of the finite-sized particles.
This is a joint work with Wolfgang König and Robert Patterson (WIAS).