Terrorists Never Congregate in Even Numbers

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.

Biography:
Andreas Kyprianou was educated at the Universities of Oxford and Sheffield. He spent two years working in the oil industry as well as holding previous academic positions at the London School of Economics, Edinburgh University, Utrecht University and at Heriot Watt University and visiting positions at Université libre de Bruxelles, ETH Zürich and CIMAT, Mexico. He came to Bath in 2006 where he co-founded and is the current director of Prob-L@B: the Probability Laboratory at Bath. In addition he is the PI and current co-director of the multi-million pound EPSRC Centre for Doctoral Training in Statistical Applied Mathematics, which is currently home to over 50 Ph.D. students.

Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai