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

We propose an index-space renormalization group (RG) approach to compare and collapse eigenvalue densities across system sizes in structured random matrix ensembles with rank-ordered power-law variance profiles. 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 illustrate this construction on generalizations of two classic random matrix ensembles-Wigner and Wishart-equipped with rank-ordered 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 index-space 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 that the scale-fixing principle underlying our RG construction extends to other structured ensembles, provided an appropriate renormalization map can be defined.