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arxiv: 2601.00220 · v2 · submitted 2026-01-01 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech· quant-ph

Anderson localisation in spatially structured random graphs

Bibek Saha , Sthitadhi Roy This is my paper

Pith reviewed 2026-05-16 18:34 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mechquant-ph
keywords Anderson localizationrandom graphslong-range hoppingphase diagraminverse participation ratioKosterlitz-Thouless scalingdisordered systemsrenormalized perturbation theory
0
0 comments X

The pith

Anderson localization vanishes beyond a critical hopping range on spatially structured graphs even at arbitrarily strong disorder.

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

The paper studies Anderson localization on random regular graphs where hopping amplitudes decay exponentially with graph distance, creating models that bridge short-range and fully connected limits. It maps the phase diagram as a function of hopping range and disorder strength using exact diagonalization and renormalized perturbation theory. Longer hopping ranges push the localization transition to higher disorder values. Past a critical range the localized phase disappears entirely, producing a direct transition between delocalized and localized states with no intervening multifractal regime. The scaling of inverse participation ratios matches Kosterlitz-Thouless-like behavior with two-parameter scaling, and average and typical correlations show distinct critical properties.

Core claim

Models are constructed by embedding a random regular graph into a complete graph and letting hopping decay exponentially with graph distance, effectively realizing power-law-like hopping generalizations of the Anderson model. Numerical and analytical results show that increasing the hopping range shifts the localization transition to stronger disorder; beyond a critical range the localized phase ceases to exist at any disorder strength. A direct Anderson transition occurs between delocalized and localized phases with no evidence for an intervening multifractal phase in both deterministic and random hopping cases, accompanied by Kosterlitz-Thouless-like scaling of inverse participation ratios

What carries the argument

Distance-dependent exponential hopping on embedded random regular graphs, diagnosed through inverse participation ratios and renormalized perturbation theory.

If this is right

  • The localization transition moves to stronger disorder with increasing hopping range.
  • Beyond the critical range no localized phase exists at any disorder strength.
  • The transition is direct with no multifractal intermediate phase for both deterministic and random hopping.
  • Inverse participation ratio scaling follows Kosterlitz-Thouless-like two-parameter behavior.
  • Average and typical correlation functions exhibit distinct critical scaling.

Where Pith is reading between the lines

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

  • Similar range-dependent suppression of localization may appear in other high-dimensional or long-range interacting disordered systems.
  • The absence of a multifractal regime could be checked by examining higher moments of wave-function distributions in larger-scale simulations.
  • The two-parameter scaling form suggests possible connections to surface or boundary criticality in related models.
  • Experimental platforms with tunable long-range couplings, such as trapped ions or Rydberg arrays, could test the predicted critical range.

Load-bearing premise

Finite-size exact diagonalization on finite graphs extrapolates reliably to the thermodynamic limit on infinite graphs without missing subtle phases or artifacts.

What would settle it

Observation of a localized phase persisting at arbitrarily strong disorder for hopping ranges larger than the reported critical value, in the limit of increasing system size.

Figures

Figures reproduced from arXiv: 2601.00220 by Bibek Saha, Sthitadhi Roy.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of the ExpRRG model. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Level statistics for the ExpRRG model. The top [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Phase diagram of the ExpRRG model obtained nu [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The generalised IPRs, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The average and typical correlation functions, de [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The scaling of the [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Results for the numerical scaling theory of the Anderson transitions in the ExpRRG model using the scaling of the [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The phase diagram of the ExpRRG model as ob [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Numerical results for the ExpRRG-RH model along [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Left: Weak link in a randomly-weighted Erd˝os-R´enyi [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
read the original abstract

We study Anderson localisation on high-dimensional graphs with spatial structure induced by long-ranged but distance-dependent hopping. To this end, we introduce a class of models that interpolate between the short-range Anderson model on a random regular graph and fully connected models with statistically uniform hopping, by embedding a random regular graph into a complete graph and allowing hopping amplitudes to decay exponentially with graph distance. The competition between the exponentially growing number of neighbours with graph distance and the exponentially decaying hopping amplitude positions our models effectively as power-law hopping generalisation of the Anderson model on random regular graphs. Using a combination of numerical exact diagonalisation and analytical renormalised perturbation theory, we establish the resulting localisation phase diagram emerging from the interplay of the lengthscale associated to the hopping range and the onsite disorder strength. We find that increasing the hopping range shifts the localisation transition to stronger disorder, and that beyond a critical range the localised phase ceases to exist even at arbitrarily strong disorder. Our results indicate a direct Anderson transition between delocalised and localised phases, with no evidence for an intervening multifractal phase, for both deterministic and random hopping models. A scaling analysis based on inverse participation ratios reveals behaviour consistent with a Kosterlitz-Thouless-like transition with two-parameter scaling, in line with Anderson transitions on high-dimensional graphs. We also observe distinct critical behaviour in average and typical correlation functions, reflecting the different scaling properties of generalised inverse participation ratios.

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 / 2 minor

Summary. The paper introduces a family of Anderson models on random regular graphs embedded in complete graphs, with hopping amplitudes decaying exponentially with graph distance. This interpolates between short-range RRG Anderson models and fully connected limits. Using exact diagonalization and renormalized perturbation theory, the authors map the localization phase diagram as a function of hopping decay length and disorder strength W. They claim that beyond a critical hopping range the localized phase is absent even at arbitrarily large W, that the transition is direct between delocalized and localized phases with no intervening multifractal regime, and that inverse-participation-ratio scaling is consistent with Kosterlitz-Thouless-like two-parameter scaling.

Significance. If the central claims survive scrutiny, the work would be significant for clarifying the role of spatial structure in high-dimensional localization problems. It provides a concrete interpolation between known limits, reports a critical hopping range that eliminates the localized phase, and identifies KT-like scaling together with distinct average/typical correlation behavior. The combination of numerical exact diagonalization and analytical renormalized perturbation theory is a strength, as is the focus on falsifiable predictions for the phase boundary.

major comments (2)
  1. [Numerical results and scaling analysis] The headline claim that the localized phase disappears beyond a critical hopping range even at arbitrarily strong disorder rests on finite-N exact diagonalization. No system sizes, range of N, functional form of the finite-size collapse, or explicit check that typical IPR remains O(1) (rather than drifting to zero) at fixed large W are reported. Given that effective coordination grows exponentially with distance on these graphs, the Thouless length can exceed the simulated diameter, making delocalization appear as an artifact rather than a thermodynamic-limit result. This directly affects the central phase-diagram conclusion.
  2. [Phase diagram and critical behavior] The assertion of a direct Anderson transition with no intervening multifractal phase is supported only by the absence of certain IPR scaling signatures. However, the manuscript does not present a systematic scan of generalized IPR exponents or participation-ratio distributions across the claimed critical line to rule out a narrow multifractal window, especially near the reported critical hopping range.
minor comments (2)
  1. [Scaling analysis] The abstract and main text refer to 'KT-like two-parameter scaling' but do not specify the two scaling variables or show the corresponding data collapse explicitly.
  2. [Methods] Error bars, convergence checks with respect to disorder realizations, and the precise definition of the renormalized perturbation theory cutoff are not provided.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major points below and will incorporate revisions to strengthen the presentation of our numerical and scaling results.

read point-by-point responses

  1. Referee: [Numerical results and scaling analysis] The headline claim that the localized phase disappears beyond a critical hopping range even at arbitrarily strong disorder rests on finite-N exact diagonalization. No system sizes, range of N, functional form of the finite-size collapse, or explicit check that typical IPR remains O(1) (rather than drifting to zero) at fixed large W are reported. Given that effective coordination grows exponentially with distance on these graphs, the Thouless length can exceed the simulated diameter, making delocalization appear as an artifact rather than a thermodynamic-limit result. This directly affects the central phase-diagram conclusion.

    Authors: We agree that additional details on the numerical implementation are needed for full clarity. In the revised manuscript we will explicitly report the system sizes employed in exact diagonalization, the range of N, the functional form used for finite-size scaling collapse of the IPR, and direct plots confirming that the typical IPR saturates to a nonzero value at fixed large W for hopping ranges above the critical value. On the Thouless-length concern, the renormalized perturbation theory supplies an independent thermodynamic-limit argument showing that the effective coordination and hopping range eliminate the localized phase beyond the critical range; the numerical data remain consistent with this prediction, and we will add an estimate of the Thouless length relative to graph diameter to address possible finite-size artifacts. revision: yes

  2. Referee: [Phase diagram and critical behavior] The assertion of a direct Anderson transition with no intervening multifractal phase is supported only by the absence of certain IPR scaling signatures. However, the manuscript does not present a systematic scan of generalized IPR exponents or participation-ratio distributions across the claimed critical line to rule out a narrow multifractal window, especially near the reported critical hopping range.

    Authors: We acknowledge that a more exhaustive check would strengthen the claim. While the existing IPR scaling and participation-ratio distributions show no multifractal signatures, we will add in the revision a systematic scan of generalized IPR exponents (D_q for several q) and participation-ratio distributions evaluated along the critical line for multiple hopping ranges, including near the reported critical value. This additional analysis, together with the renormalized perturbation theory, will help confirm the absence of an intervening multifractal window. revision: yes

Circularity Check

0 steps flagged

No circularity: independent numerical and analytical checks on graph-defined model

full rationale

The derivation combines exact diagonalization on finite graphs with renormalized perturbation theory applied to the same distance-dependent hopping model. Model parameters and graph embedding are introduced directly from the construction without any fitted quantity being relabeled as a prediction. No self-citation is load-bearing for the central claim of a direct Anderson transition beyond a critical range; the numerical spectra serve as an external benchmark. The KT-like scaling analysis is presented as consistency check rather than a derived necessity. The result remains falsifiable by larger-system simulations or alternative methods.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the models introduce tunable physical parameters for hopping decay and disorder but no new fundamental entities. Axioms are standard assumptions of the Anderson model and graph theory.

free parameters (2)
  • hopping decay length
    Parameter controlling exponential decay of hopping amplitudes with graph distance, used to interpolate between model limits.
  • disorder strength W
    Onsite potential disorder strength varied to determine the localisation transition.
axioms (2)
  • domain assumption Random regular graphs admit a well-defined graph distance when embedded in a complete graph.
    Foundation for constructing the spatially structured hopping.
  • domain assumption Renormalised perturbation theory is applicable to the localisation problem on these graphs.
    Basis for the analytical component of the phase diagram.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Anderson Localization on Husimi Trees and its implications for Many-Body localization

    cond-mat.dis-nn 2026-01 unverdicted novelty 5.0

    Local loops on Husimi trees reduce the critical disorder for Anderson localization and increase the spatial extent of localized eigenstates, providing a better single-particle analogy for many-body localization.

Reference graph

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