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arxiv: 2606.10915 · v1 · pith:SW7BHCFNnew · submitted 2026-06-09 · ❄️ cond-mat.dis-nn

Local density of states distribution and multifractal eigenvectors of weighted random networks via the cavity approach

Joseph W. Baron , Tim Rogers This is my paper

Pith reviewed 2026-06-27 10:54 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords local density of statesmultifractal eigenvectorscavity methodweighted Erdős-Rényi graphsinverse participation ratiopower-law tailsextended phasesingularity spectrum
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The pith

The local density of states on weighted random networks has power-law tails of exponent 3, implying weakly multifractal eigenvectors with logarithmic inverse participation ratio scaling.

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

The paper uses the cavity method to approximate the full distribution of local density of states on weighted Erdős-Rényi graphs. It derives compact expressions showing that the tails of this distribution follow a power law with exponent 3 throughout the extended phase. From this the authors conclude that eigenvectors belonging to the continuous spectrum are extended yet weakly multifractal, and they obtain expressions for the associated fractal dimensions and singularity spectrum. They further show that the inverse participation ratio in this regime scales logarithmically with system size, which is neither the constant value of fully extended states nor the power-law decay of localized states. The work also verifies that symmetry properties previously derived from the non-linear sigma model continue to hold for both the local density of states distribution and the singularity spectrum.

Core claim

Using the cavity approach on a general class of weighted Erdős-Rényi graphs, we obtain a good approximation to the full local density of states distribution together with compact expressions for its power-law tails, which have exponent 3 in the extended phase. We deduce that the eigenvectors in the continuous part of the spectrum are extended but weakly multifractal, extract the associated fractal dimensions and singularity spectrum, and demonstrate that the inverse participation ratio exhibits an unusual logarithmic scaling with system size. Finally we verify that symmetry properties derived from the non-linear sigma model hold for both the local density of states distribution and the singu

What carries the argument

The cavity method applied to the local density of states on weighted Erdős-Rényi graphs, which produces the full distribution and its power-law tails.

If this is right

  • Eigenvectors in the continuous spectrum are extended but weakly multifractal.
  • The inverse participation ratio scales logarithmically with system size rather than remaining constant or decaying as a power law.
  • The fractal dimensions and singularity spectrum follow from the power-law tails of the local density of states.
  • Symmetry properties from the non-linear sigma model hold for the local density of states distribution and the singularity spectrum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same cavity-method route to the local density of states distribution could be applied to other sparse weighted graph ensembles to test whether exponent-3 tails are generic.
  • Finite-size scaling studies on graphs of increasing size could directly check whether the logarithmic inverse participation ratio persists before the thermodynamic limit is reached.
  • The observed multifractality may provide a concrete example for testing general predictions about critical states at the localization transition on networks.

Load-bearing premise

The cavity method supplies a quantitatively accurate approximation to the full local density of states distribution and its tails throughout the extended phase in the thermodynamic limit.

What would settle it

Numerical diagonalization of finite but large weighted Erdős-Rényi graphs that yields a local density of states tail exponent measurably different from 3, or an inverse participation ratio that fails to scale logarithmically with system size, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.10915 by Joseph W. Baron, Tim Rogers.

Figure 1
Figure 1. Figure 1: FIG. 1: Global density of states [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Local density of states distribution for the ER graph [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Singularity spectrum for the three weight [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Logarithmic scaling of the IPR with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We study the local density of states (LDoS) distribution of a general class of weighted Erd\H{o}s-R\'enyi graphs. Using the cavity method, we obtain a good approximation to the full LDoS distribution and compact expressions for its power-law tails, which we show to have exponent $3$ in the extended phase. We deduce that the eigenvectors in the continuous part of the spectrum are extended but (weakly) multifractal, and we extract expressions for the associated fractal dimensions and the singularity spectrum. We also demonstrate that the inverse participation ratio in this multifractal phase exhibits an unusual logarithmic scaling with system size, which is neither fully-extended nor localised by the usual definitions. Finally, we verify that some symmetry properties (derived from the non-linear sigma model), which have been shown to hold for many systems exhibiting multifractality, also hold in our case, both for the LDoS distribution and the singularity spectrum.

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

0 major / 3 minor

Summary. The manuscript applies the cavity method to a class of weighted Erdős-Rényi graphs to obtain an approximation to the full local density of states (LDoS) distribution. It derives compact expressions for the power-law tails of this distribution (exponent 3 in the extended phase), deduces that eigenvectors in the continuous spectrum are extended but weakly multifractal, extracts the associated fractal dimensions and singularity spectrum, demonstrates logarithmic scaling of the inverse participation ratio with system size, and verifies that non-linear sigma model symmetry properties hold for both the LDoS distribution and the singularity spectrum.

Significance. If the derivations hold, the work supplies an analytically tractable route to the full LDoS distribution and its multifractal consequences on locally tree-like graphs, where the cavity method is exact in the thermodynamic limit. The explicit large-argument asymptotics yielding the exponent-3 tails, the resulting weak multifractality with logarithmic IPR scaling, and the independent consistency check against NLSM symmetries constitute clear strengths. These results are relevant to localization and spectral statistics in disordered networks.

minor comments (3)
  1. [cavity equations section] § on cavity equations: the transition from the distributional fixed-point equations to the explicit tail asymptotics should include one intermediate step showing how the exponent 3 emerges from the large-argument behavior without additional assumptions.
  2. [figures] Figure captions for the numerical checks of the LDoS tails and IPR scaling should state the system sizes and number of realizations used, to allow direct assessment of finite-size effects.
  3. [model definition] Notation for the weighted adjacency matrix and the cavity fields should be introduced once with a single consistent symbol set to avoid minor ambiguity in later sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation applies the cavity method (standard for locally tree-like ER graphs) to obtain self-consistent distributional equations for the LDoS. Asymptotics of these equations directly yield the power-law tails of exponent 3, from which the weak multifractality, fractal dimensions, and logarithmic IPR scaling follow via moment relations. No parameters are fitted to the target observables, no self-citations bear the central claims, and no ansatz or uniqueness theorem is smuggled in; the NLSM symmetry checks supply independent consistency support. The result is self-contained against the cavity equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central results rest on the applicability of the cavity method to the thermodynamic limit of sparse weighted graphs and on the validity of the non-linear sigma model symmetries as external benchmarks.

axioms (2)
  • domain assumption The cavity method yields a quantitatively accurate approximation to the LDoS distribution in the thermodynamic limit.
    Invoked throughout the abstract as the source of the distribution and tail expressions.
  • domain assumption Symmetry properties derived from the non-linear sigma model hold for the LDoS distribution and singularity spectrum.
    Used as an external check whose agreement is presented as supporting evidence.

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Reference graph

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