Renormalization group for spectral collapse in random matrices with power-law variance profiles

Abstract

We propose a renormalization group (RG) approach to compare and collapse eigenvalue densities of random matrix models of complex systems across different system sizes. The approach is to fix a natural spectral scale by letting the model normalization run with size, turning raw spectra into comparable, collapsed density curves. We demonstrate this approach on generalizations of two classic random matrix ensembles--Wigner and Wishart--modified to have power-law variance profiles. We use random matrix theory methods to derive self-consistent fixed-point equations for the resolvent to compute their eigenvalue densities, we define an RG scheme based on matrix decimation, and compute the Beta function controlling the RG flow as a function of the variance profile power-law exponent. The running normalization leads to spectral collapse which we confirm in simulations and solutions of the fixed-point equations. We expect this RG approach to carry over to other ensembles, providing a method for data analysis of a broad range of complex systems.