Universality in Complexity: A Generalized Random Matrix Approach

Physical systems can be complicated. Lack of detailed knowledge of their interactions can be represented by randomness in generators of the dynamics. In waves (quantum, electromagnetic, etc.) the associated operators can be modelled by random matrices.

The choice of a suitable random-matrix model is sensitive to the nature of the complexity. The statistical analysis therefore requires exploration of a wide range of random-matrix ensembles, with the aim of identifying and analyzing common mathematical structures among their statistics. Our search leads to to a common framework: single-parametric Brownian ensembles.

This common mathematical framework suggests an analogous formulation for physical properties, and their classification into universality classes defined by the single (complexity) parameter. The analogy helps to reveal many important features of the level statistics in interacting systems e.g. a critical point different from that of non-interacting systems, the possibility of extended states even in one dimension. This further suggests a deep-rooted universality hidden underneath the world of complex systems.