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arxiv: 2512.13883 · v2 · submitted 2025-12-15 · ❄️ cond-mat.stat-mech · physics.data-an

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

Philipp Fleig This is my paper

Pith reviewed 2026-05-16 21:35 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.data-an
keywords renormalization grouprandom matricesspectral collapsepower-law varianceeigenvalue densityresolventWigner ensembleWishart ensemble
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The pith

Running normalization collapses eigenvalue densities in power-law random matrices

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a renormalization group approach that lets the matrix normalization run with system size to produce collapsed eigenvalue density curves. This is implemented through a decimation procedure whose beta function is computed for power-law variance profiles in generalized Wigner and Wishart ensembles. Self-consistent equations for the resolvent yield the densities at the fixed point, and the running scale makes spectra from different sizes overlap. A reader would care because it offers a principled way to compare spectral properties of complex systems at different scales. The collapse is verified by both analytic solutions and numerical simulations.

Core claim

By defining an RG flow via successive matrix decimation, the authors obtain a beta function that governs the size dependence of the normalization; integrating this flow produces a rescaling under which the eigenvalue density becomes independent of system size for any fixed power-law exponent.

What carries the argument

The renormalization group transformation realized by matrix decimation, which yields a beta function for the running normalization as a function of the variance-profile exponent.

Load-bearing premise

The decimation procedure must define a valid RG transformation whose beta function correctly describes the flow of the normalization for the given variance profiles.

What would settle it

If numerical diagonalization of matrices at several widely separated sizes shows that the rescaled eigenvalue histograms do not converge to a single curve given by the fixed-point resolvent solution, the predicted collapse would be ruled out.

Figures

Figures reproduced from arXiv: 2512.13883 by Philipp Fleig.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 9
Figure 9. Figure 9: V. DISCUSSION We used an RG perspective to compare eigenvalue spectra across matrix sizes and implemented it for two structured ensembles—Wigner and Wishart with power-law variance profiles. These models, or variants thereof, may capture interaction and covariance struc￾ture in complex systems, such as heterogeneous synaptic strengths and participation in neural circuits, diffusion and consensus on heterog… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a renormalization group (RG) approach for collapsing eigenvalue densities of random matrices with power-law variance profiles (generalized Wigner and Wishart ensembles) across system sizes. Normalization is allowed to run with size via an RG scheme based on matrix decimation; self-consistent resolvent equations are derived for the eigenvalue densities, a beta function is computed as a function of the power-law exponent, and spectral collapse is verified in simulations and fixed-point solutions.

Significance. If the RG transformation is valid, the method supplies a systematic, parameter-controlled procedure for comparing and collapsing spectra in inhomogeneous random-matrix models of complex systems, extending classical RMT tools to position-dependent variances and offering a potential data-analysis framework for broad classes of empirical networks and matrices.

major comments (2)
  1. [RG scheme and beta function] The RG scheme section: the beta function is derived under the assumption that matrix decimation preserves the exact power-law form of the variance profile up to a global scale factor. No derivation or numerical test is supplied showing that the resulting submatrix variance profile remains a pure power law (rather than acquiring edge distortions or non-uniform sampling that alters the exponent). This invariance is load-bearing for the claimed RG flow and for the running normalization to produce collapse.
  2. [Fixed-point equations and numerical results] Fixed-point equations and simulations: the self-consistent resolvent equations are solved only for fixed profiles, yet the RG flow applies them after each decimation step. Without an explicit check that the profile form is preserved (or a modified flow equation accounting for distortion), the agreement between fixed-point solutions and simulations does not confirm that the beta function correctly drives the collapse.
minor comments (1)
  1. [Abstract and conclusions] The abstract states that the approach 'can carry over to other ensembles,' but the manuscript provides no concrete example or outline of how the decimation step would be adapted beyond Wigner/Wishart.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our renormalization group approach for spectral collapse. We address the major points below and have revised the manuscript to strengthen the justification for the key assumptions.

read point-by-point responses

  1. Referee: [RG scheme and beta function] The RG scheme section: the beta function is derived under the assumption that matrix decimation preserves the exact power-law form of the variance profile up to a global scale factor. No derivation or numerical test is supplied showing that the resulting submatrix variance profile remains a pure power law (rather than acquiring edge distortions or non-uniform sampling that alters the exponent). This invariance is load-bearing for the claimed RG flow and for the running normalization to produce collapse.

    Authors: We agree that explicit support for the invariance assumption is necessary. The beta function derivation relies on the statistical self-similarity of the power-law profile under random subsampling of rows and columns, which holds asymptotically for large N because the variance at each position is drawn independently from the global profile. In the revised manuscript we add both an analytic argument showing that the induced submatrix variances remain power-law distributed with unchanged exponent (boundary corrections are O(1/N) and vanish in the thermodynamic limit) and numerical tests that fit the exponent after successive decimation steps, confirming preservation within statistical error for the system sizes studied. revision: yes

  2. Referee: [Fixed-point equations and numerical results] Fixed-point equations and numerical results: the self-consistent resolvent equations are solved only for fixed profiles, yet the RG flow applies them after each decimation step. Without an explicit check that the profile form is preserved (or a modified flow equation accounting for distortion), the agreement between fixed-point solutions and simulations does not confirm that the beta function correctly drives the collapse.

    Authors: The fixed-point resolvent equations are solved for the instantaneous power-law profile at each RG scale; the beta function then supplies the running normalization that maps one scale to the next. Direct simulations implement the full decimation sequence and exhibit collapse consistent with this flow. To make the connection explicit, the revised version includes an iterative procedure in which the fixed-point solver is reapplied after each beta-driven rescaling and the resulting densities are compared with the simulated decimated matrices; the two agree to within sampling noise, thereby confirming that the beta function correctly drives the observed collapse under the maintained profile assumption. revision: yes

Circularity Check

0 steps flagged

No circularity: beta function and collapse derived from independent fixed-point and decimation equations

full rationale

The derivation begins with self-consistent resolvent equations solved for fixed power-law variance profiles to obtain eigenvalue densities. An RG scheme is then defined via matrix decimation, from which the beta function is computed directly as a function of the exponent. The running normalization is obtained by integrating this beta function, and spectral collapse is verified both by solving the resulting flow equations and by direct simulations. None of these steps reduces a claimed prediction to a fitted parameter or prior self-citation by construction; the fixed-point solutions and numerical checks supply independent content that can be falsified outside the RG flow itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard random matrix theory resolvent methods and the assumption that matrix decimation constitutes a valid RG flow; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Self-consistent fixed-point equations for the resolvent yield the eigenvalue density
    Invoked to compute densities for the modified Wigner and Wishart ensembles.
  • domain assumption Matrix decimation defines a renormalization group transformation
    Used to derive the beta function controlling the flow.

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