Condensation in large random Lotka-Volterra systems

凝聚是科学中的一种普遍现象，可以在大的相互作用系统中发现，每个相互作用系统都可能以不同的状态存在。如果这些系统中的有限部分选择处于相同的状态，就会发生凝聚。例子从原子气体中的玻色-爱因斯坦凝聚到生物物种的出现。然而，在平衡情况下，凝聚态往往是唯一的，非平衡稳态通常表现出许多不同的凝聚态模式，凝聚态的数目和选择成为相关问题。研究这些问题的一个合适的数学框架是所谓的复制因子方程。它描述了处于不同状态的系统分数的时间演化。这个方程最初是在进化博弈论领域引入的，最近被证明可以对发生凝结的各种情况进行建模。它的主要成分是一个相互作用矩阵，其中包含不同状态之间可能发生的随机转换的速率。现实情况的特点往往是大量的状态和复杂的交互。为了理解在这种条件下凝结的典型特征，合理地假设相互作用矩阵是从随机矩阵集合中提取的。如果凝聚过程的关键性质与相互作用矩阵的具体实现无关，而仅取决于整个系综的统计性质，那么这种方法将是卓有成效的。最近在数值模拟中发现了沿着这些推理路线的几个有趣的结果。在本项目中，我们希望使用无序系统统计力学的方法来补充和扩展这些数值结果，通过分析凝聚态的数量分布、它对基本相互作用网络的连接性的依赖以及所选择的凝聚态图案的稳定性。

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.