Universality in complexity: a random matrix view-point

For systems that can be described mathematically, physical information can be derived, in principle, from detailed knowledge of the operators that govern their evolution. Physical systems can however be complex in nature and it is not always possible to determine the operator exactly or, even if they are known, to solve the equations they determine.  This paper aims to model the statistical behaviour of those complex systems where a matrix representation of the operators is meaningful.

The complexity of a system, in general, makes it difficult to determine some or almost all matrix elements of its operators. The lack of accuracy acts as a source of randomness for the matrix elements which are also subjected to an external potential due to existing system conditions.  The fluctuation of accuracy due to varying system conditions leads to a diffusion of the matrix elements. We show that, for single-well potentials, the diffusion can be described by a common mathematical formulation where system information enters through a single parameter. This suggests a possible classification of complex systems in an infinite range of universality classes characterized just by a single parameter and the nature of global physical constraints. It seems to indicate a web of connection hidden underneath complex systems.