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

Area of Science:

• * Statistical Physics
• * Condensed Matter Physics
• * Random Matrix Theory

Background:

• * Analyzing eigenvalue densities in large random matrices is crucial for understanding complex systems.
• * Traditional methods struggle to compare spectra across varying system sizes.
• * Structured random matrix ensembles with power-law variance profiles present unique analytical challenges.

Purpose of the Study:

• * To introduce a novel index-space renormalization group (RG) approach for comparing and collapsing eigenvalue densities.
• * To apply this RG method to generalized Wigner and Wishart random matrix ensembles with specific variance profiles.
• * To establish a scale-fixing principle for spectral analysis in structured random matrices.

Main Methods:

• * Developed an index-space renormalization group (RG) approach based on decimation.
• * Utilized random matrix theory to derive self-consistent fixed-point equations for the resolvent.
• * Computed the Beta function to characterize the RG flow concerning the variance profile exponent.

Main Results:

• * Demonstrated that a running model normalization fixes a natural spectral scale, enabling data collapse.
• * Successfully computed eigenvalue densities for generalized Wigner and Wishart ensembles.
• * Confirmed spectral collapse through numerical simulations and analytical solutions.

Conclusions:

• * The proposed RG approach effectively collapses eigenvalue densities across system sizes in structured random matrices.
• * The scale-fixing principle is a powerful tool for analyzing spectral properties in complex matrix ensembles.
• * This method is expected to be applicable to other structured ensembles with a defined renormalization map.