Asymptotic Properties of Complex Random Systems and Applications
M. Luczak (London School of Economics)
In recent years there has been a surge of interest in the mathematics of real-world networks, such as the Internet, telecommunications networks, social networks, biological systems (e.g. populations, ecosystems), chemical reactions (e.g coagulation processes). These can often be represented by combinatorial and stochastic models, e.g random graph processes modelling evolving web graphs, or Markov processes modelling the spread of epidemics). Under certain conditions, there is a law of large numbers, i.e. the dynamics of a complex interacting system can be approximated by a deterministic process solving a differential equation derived from the average drift, with much simpler dynamics. If also fluctuations converge, then a central limit theorem holds. Further, often one may observe chaoticity, i.e. asymptotic independence of particles. In essence, such results mean that one can understand the behaviour of a complex system via that of a simpler system. Unfortunately, proving such approximations is often difficult, especially when the random processes can have an unbounded number of components. At the moment, it seems that each new problem defies existing approaches in an interesting way, and we lack a coherent and widely applicable theory. We hope that over the coming years, the intense activity in this area will produce such a theory, which will have important implications for industry.