Condensation in large random Lotka-Volterra systems

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Condensation is a general phenomenon in science to be found in large sets of interacting systems each of which may exist in different states. Condensation is said to occur if a finite fraction of these systems choose to be in the same state. Examples reach from Bose-Einstein condensation in atomic gases to the emergence of biological species. Whereas in equilibrium situations the condensed state is often unique non-equilibrium steady states typically show many different condensation patterns and the question of the number and selection of condensed states becomes relevant. A suitable mathematical framework to investigate these questions is the so-called replicator equation. It describes the time evolution of the fraction of systems in the various states. Being introduced originally in the field of evolutionary game theory this equation has been recently shown to model a variety of situations in which condensation occurs. Its main ingredient is an interaction matrix containing the rates of possible stochastic transitions between different states. Realistic situations are often characterized by a large number of states and complex interactions. To understand typical features of condensation under such conditions it is reasonable to assume that the interaction matrix is drawn from a random matrix ensemble. Such an approach will be fruitful if crucial properties of the condensation process are independent of the concrete realization of the interaction matrix and depend only on the statistical properties of the whole ensemble. Several interesting results along those lines of reasoning have been found recently in numerical simulations. In the present project we want to employ methods from the statistical mechanics of disordered systems to complement and extend these numerical findings by analytical results on the distribution of the number of condensates, its dependence on the connectivity of the underlying interaction network, and the stability of the selected pattern of condensed states.