SEMINARIO DI CALCOLO DELLE PROBABILITA'
Ho il piacere di invitarvi al seguente seminario, che si terrà Venerdì
12 Ottobre alle 11:30, nella Sala Riunioni del VII piano della Torre
Archimede, in Via Trieste 63, Padova,
LUISA ANDREIS (Weierstraß-Institut für Angewandte Analysis und
Stochastik - WIAS)
Titolo: A large-deviations approach to the multiplicative coagulation
process
Abstract: 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).
Paolo Dai Pra