Koç University, Mathematics Seminar
Date & Time: Thursday, May 5, 16:00-17:00
Place: SCI 129
Speaker: Andreas Kyprianou (University of Bath)
Title: : Some strange results in fragmentation-coalescence
models
Abstract: We analyse a class of fragmentation-coalescence
processes defined on finite systems of particles organised into clusters.
Coalescent events merge multiple clusters simultaneously to form a single
larger cluster, while fragmentation breaks up a cluster into a collection of
singletons. Under mild conditions on the coalescence rates, we show that the
distribution of cluster sizes becomes non-random in the large-scale limit.
Moreover, we discover that, in the limit of small fragmentation rate, these
processes exhibit a universal heavy tailed distribution with exponent 3/2. In
addition, we observe a strange phenomenon that if coalescence of clusters
always involves 3 or more blocks, then the large-scale limit has no even sided
blocks. Some complementary results are also presented for exchangeable
fragmentation-coalescence processes on partitions of natural numbers. In this
case one may work directly with the infinite system and we ask whether the
process can come down from infinity. The answer reveals a remarkable dichotomy.
This is based on two different pieces of work with Tim
Rogers, Steven Pagett and Jason Schweinsberg.
Speaker's
Brief Biography: Andreas E. Kyprianou is the
co-director of Prob-L@B in Department of Mathematical Sciences at the
University of Bath. He has contributed to Probability Theory with numerous
publications based on his expertise on a wide range of topics. His research
interests include branching processes, fixed point equations and travelling
waves, random walks, Brownian motion, Lévy processes, stochastic games and
stochastic control problems.