Minisymposium Asymptotic Properties of Complex Random Systems and Applications

Abstract

There has been recent intense activity in the study of the asymptotic character of sequences of random processes arising e.g. in computer science, statistical physics and mathematical biology. These may model the emergence of certain graph properties; load-sharing among links or servers; the survival and extinction of species; co-operation and competition in a social context; spread of epidemics; DNA, RNA and amino-acid sequences. Under appropriate conditions,a sequence of processes converges to the solution of a differential equation, which may be interpreted as a functional law of large numbers. Such approximations are of great significance as a way to interpret the qualitative behaviour of a complicated, multi-faceted structure in terms of a considerably simpler one. Unfortunately, it is often difficult to prove their validity, especially when the random process has an unbounded number of components in the limit. We would hope that over the coming years, the intense interest in the field will produce a coherent and widely applicable theory. At present, it often appears that each new problem defies the existing theory in an interesting way. In organising the ECMI Minisymposium `Asymptotic properties of complex random systems and applications´, our motivation was to bring these problems into focus and highlight their importance in modelling of real-world situations. Our aim was thus to generate interest among the applied mathematics community, in the hope that interesting new insights and ideas may result. The following sections summarise the contents of the four talks given.