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

TL;DR

The paper develops an RG framework to collapse and compare eigenvalue densities of random matrices across system sizes by enforcing a fixed spectral scale via a running normalization. It derives self‑consistent resolvent equations for Wigner and Wishart ensembles with power‑law variance profiles, identifies a beta function β_RG = 1 − 2α that governs the RG flow under decimation, and demonstrates spectral collapse consistent with Callan–Symanzik invariance in the large‑N limit. The analysis reveals distinct RG regimes: a relevant regime for α < 1/2, a marginal regime at α = 1/2 with nontrivial fixed points (Wigner) or logarithmic corrections (Wishart), and an irrelevant regime for α > 1/2 where the bulk collapses toward zero while outliers persist. The results provide a robust, scale‑aware method for analyzing spectral data from heterogeneous or high‑dimensional systems and suggest broader applicability to other ensembles and data analysis tasks, with a clear path for incorporating finite‑N corrections and universality tests.

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.

Figures (9)

• Figure 1: Eigenvalue densities of Wigner ensemble with power-law variance profile. (a) and (b) Eigenvalue density for different values of $\alpha$ from simulations (gray) and fixed-point solutions of the self-consistent finite-$N$ equations (red). Across all plots we use $\gamma=1$, $N=600$, $\sigma^2=1$, $20$ independent simulation trials, and $\alpha$ values as indicated on the subplots. All curves are Cauchy-smoothed with kernel width $\eta=10^{-2}$. Linear-linear scale (a) and log-linear scale (b).
• Figure 2: Eigenvalue density collapse and running normalization. (a) and (b) Raw and collapsed eigenvalue densities obtained from simulations of the ensemble. The raw densities (left subplot) use $\gamma=1$ and the collapsed densities are obtained using the running normalization $\gamma(N)$. We use $\alpha=0.3$ and $N=200$, $800$, and $2000$ and ten independent ensemble trials. For $N=200$ we also show the one standard deviation band of sampling noise. Linear-linear scale (a) and log-linear scale (b). (c) Decimation flow of normalization: $\gamma(K)$ versus retained size $K$ after removing a fixed fraction of largest indices at each step, while the second moment is kept fixed at unity. Solid lines: exact prediction. Markers with error bars: simulation with mean and standard deviation from six independent trials. Dashed lines serve as guides indicating slope $1-2\alpha$. Dashed lines are mostly covered by the exact prediction. (d) Decimation flow of the coupling $c$.
• Figure 3: Spectral collapse for the finite-$N$ self-consistent equation. Spectral invariance test for $\alpha=0.30$ across $N=200$, $800$, and $2000$. Left: raw eigenvalue density $\rho_{N,\eta}(x)$ with $\gamma=1$ computed from the finite-$N$ self-consistent equation for different $N$. Right: collapsed eigenvalue density with running normalization $\gamma(N)$. We use $\eta_\mathrm{raw}=\sqrt{\gamma(N)}\eta_\mathrm{coll}$, with $\eta_\mathrm{coll}=10^{-2}$ for all $N$.
• Figure 4: Wishart ensemble with power-law variance profile. (a) Comparison of spectral distribution simulations (gray histograms) and the fixed-point solution of the self-consistent equations (red). Across all subplots we use $\gamma=1$, $N=200$, $T=400$ (i.e. $q=0.5$), $\sigma^2=1$, and $\eta=10^{-2}$, with $\alpha$ values as indicated above the plots. (b) Raw and collapsed eigenvalue densities obtained from simulations of the ensemble. We use $\alpha=0.3$ and $N=200$, $800$, and $2000$, and $q=0.5$ and simulate $100$ independent trials of the ensemble. (c) Raw and collapsed eigenvalue density from the fixed-point solution for the same parameters as (b). Left: raw densities with $\eta_\mathrm{raw}=\gamma^2(T)\, \eta_\mathrm{coll}$. Right: collapsed densities with $\eta_\mathrm{coll}=10^{-2}$ for all $N$.
• Figure 5: Finite-size scaling of the second moment for representative exponents $\alpha$. Solid lines: exact formula in Eq. \ref{['eq:second_moment']}. Markers: Simulation estimates from eight independent trials. Dashed guides: predicted slopes.