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REVIEW 3 major objections 5 minor 82 references

Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices

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Pith's one-line read The paper claims that the holographic boundary correlations of hyperbolic lattices survive the random insertion of Euclidean hexagon defects up to a defect fraction of about 90 percent, while the bulk density of states responds much more qu

desk verdict Clean numerical study: boundary power-laws survive high defect fraction, but the practical claim is built on ensemble averages and needs per-realization data. read the letter →

arxiv 2602.00231 v2 pith:JY2YLQ4U submitted 2026-01-30 cond-mat.mes-hall cond-mat.stat-mechhep-thmath-phmath.MP

Mireia Tolosa-Simeón , Igor Boettcher This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.stat-mechhep-thmath-phmath.MP
keywords hyperboliclatticesEuclideandefectstile-by-tileinflationboundarycorrelationsholographydensityofstatestight-bindingmodelAdS/CFTtoy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies tight-binding particles on planar graphs that interpolate from hyperbolic {7,3} or {8,3} tilings to the Euclidean {6,3} honeycomb by replacing polygons with hexagons at a controlled average fraction rho. Its central claim is that boundary observables barely notice these Euclidean defects: the logarithmic relation between graph distance and boundary distance, and the resulting power-law decay of the boundary two-point function, persist up to rho about 0.9. Bulk observables are far more sensitive, with sharp spectral peaks vanishing already near rho about 0.1 and honeycomb van Hove singularities appearing only near rho about 1. The practical upshot is that essential boundary physics associated with discrete holography can be reproduced in smaller defective lattices rather than in pure hyperbolic flakes, whose site numbers grow exponentially with graph diameter.

What carries the argument

The load-bearing construction is tile-by-tile inflation with up/down word sequences: each ring of the flake is generated by replacing every outward-pointing site with a p-gon tile, with p chosen per tile (here p=6 with probability rho and p=7 otherwise), and the adjacency matrix is read off from the up/down words. The physical argument then rides on two diagnostic identities: the distance scaling d_ab = A exp(b D_ab) for hyperbolic flakes versus d_ab = C D_ab for Euclidean ones, and the boundary two-point function reduced to the bulk propagator on the two sites adjacent to the boundary sites. The bulk density of states is computed from the same graphs but restricted to sites of coordination

What would settle it

Simulate two flakes with the same fraction of hexagons, say rho about 0.7, one with all hexagons clustered near the central polygon and one with hexagons forming a band adjacent to the boundary, then compare the boundary two-point function and the d_ab versus D_ab fits; if the power-law exponent or the logarithmic distance relation changes materially between the two arrangements, then arrangement, not rho alone, is controlling the robustness.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a separation of scales in the hyperbolic-to-Euclidean crossover: boundary-to-boundary correlations remain of hyperbolic type in the presence of a large fraction of Euclidean defects. Concretely, for each rho less than 1, the authors fit the boundary distance versus graph distance as d_ab approximately A exp(b D_ab) and find b decreasing slowly from 0.23 at rho=0 to 0.08 at rho=0.9, rather than switching to the linear Euclidean relation d_ab approximately C D_ab. The boundary two-point function, reduced to the bulk propagator between the two adjacent bulk sites, then decays as a power law 1/d_ab^(2Delta). The same flakes show a bulk density of states

Load-bearing premise

The load-bearing premise is that the hexagon fraction rho, averaged over 30 random placements, controls boundary robustness, while the paper notes that individual realizations deviate more strongly, so if a practical flake's arrangement rather than its rho decides whether power-law correlations survive, the smaller-lattice conclusion would not hold for every flake.

Editorial extensions

If this is right

  • Boundary CFT-like power laws survive in flakes where about 90 percent of the polygons are Euclidean hexagons, so experiments and numerics can use far smaller defective lattices instead of pure hyperbolic flakes.
  • The extracted scaling dimension Delta in defective flakes remains close to the hyperbolic relation m^2 L^2 approximately Delta(Delta-1), meaning the continuum AdS_2 formula is a useful estimate across most of the crossover.
  • Bulk properties are not equally protected: the sharp {7,3} spectral peak disappears by rho about 0.1, while honeycomb van Hove singularities and the linear low-energy density of states only appear for rho above about 0.95.
  • The tile-by-tile inflation scheme provides a systematic way to generate planar graphs with fixed coordination number and arbitrary polygon mixes, so the same method can tune other defect types or base curvatures.
  • The persistence of hyperbolic distance scaling up to large rho indicates that graph-geodesic compression through the bulk is the mechanism protecting boundary correlations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the robustness is probably controlled by whether graph geodesics still pass through the bulk, not by local curvature per se; a testable extension would be to replace hexagons with pentagons, which carry positive curvature, and see whether power-law boundary correlations break down at much smaller defect fractions.
  • Editorial inference: the averaging caveat suggests an optimization problem, namely arranging defects in a center-clustered or boundary-sparse pattern to minimize deviations and reduce lattice size further; the paper does not explore this.
  • Editorial inference: if the boundary power law is what underlies holographic quantum error-correcting code performance, these defective flakes could serve as lower-overhead code geometries, though the paper does not compute code distances or noise thresholds.
  • Editorial inference: the near-threshold persistence of boundary scaling hints at a possible soft boundary transition versus a sharp bulk spectral transition; measuring boundary entropy or mutual information across the crossover could clarify whether there is a genuine boundary phase transition.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies tight-binding models on planar graphs interpolating between hyperbolic {7,3}/{8,3} lattices and the Euclidean {6,3} honeycomb lattice, using the fraction ρ of randomly placed hexagonal defects. The authors analyze three observables: the relation between boundary distance d_ab and graph distance D_ab (Fig. 4), the boundary two-point correlation function ⟨O_a O_b⟩ (Fig. 5), and the bulk density of states (Figs. 7 and 8). They extract a correlation length ξ and a scaling dimension Δ by fitting the boundary correlator to exponential and power-law forms (Eqs. (23) and (24), Fig. 6). The central claim is that boundary observables are remarkably robust up to ρ ≲ 0.9, while bulk properties, especially sharp peaks in the DOS, are strongly affected; they conclude that defective lattices with fewer sites can reproduce essential boundary physics of pure hyperbolic lattices.

Significance. If the central robustness claim holds, the paper would have a clear practical payoff: experimental and numerical realizations of hyperbolic boundary physics (e.g., AdS/CFT-like holographic correlations) might not require exponentially large pure hyperbolic flakes. The construction via tile-by-tile inflation is clearly described and gives a precise, reproducible family of graphs with tuneable defect fraction. The study also usefully separates bulk and boundary observables, showing that they respond differently to the same geometric perturbations. However, the practical claim hinges on the behavior of individual defective flakes, and the manuscript currently provides only ensemble-averaged evidence for that behavior; the paper does not ship code or data, and the extracted ξ and Δ come from fits without uncertainties.

major comments (3)
  1. [§III.B, Fig. 5 and text after Fig. 5] The central robustness claim—that boundary correlations survive at high defect fraction and that ‘defective lattices with fewer sites may be sufficient’—is supported only by data averaged over 30 random defect realizations. The text explicitly states that individual realizations show more pronounced deviations and that some arrangements may minimize deviations, but no per-realization curves, spread of fitted exponents, or worst-case analysis is provided. An experiment or a finite numerical simulation uses one flake, not an ensemble average; if the spread is large, the practical conclusion does not follow. Please report the realization-to-realization distribution of the fitted ξ and Δ (or of the distance-fit parameters) at each ρ, and state how many of the 30 realizations give a clean power-law regime over at least a decade in d_ab.
  2. [§III.B, Eqs. (16), (23), (24), Fig. 5] The claimed power-law boundary behavior is extracted by fitting the distance relation to d_ab = A exp(b D_ab) and the correlator separately to exp(-D_ab/ξ) and to d_ab^{-2Δ}. This is not a parameter-free test of holographic scaling: the functional forms are assumed, and the same data are used to infer the transformation and the decay. No residuals or goodness-of-fit metrics are shown for the fits in Fig. 5, and Fig. 6 reports ξ and Δ without error bars. Given that the paper’s main claim is robustness of the power law, the analysis must show either that the power-law fit is significantly better than, say, an exponential in d_ab, or provide fit uncertainties and residuals, especially at ρ = 0.90 where deviations are visible at large d_ab.
  3. [§II, §III.C, Figs. 7, 8 and Table I] The defect-ensemble construction is controlled by the hexagon fraction ρ, but different realizations at the same ρ have different ring counts l and different total sizes N (Table I), and the DOS curves in Figs. 7 and 8 are computed from 30-realization averages with no uncertainty band. The DOS conclusions, such as ‘sharp peak around E ≃ 3 is eliminated for ρ ∼ 0.1’ for the {7,3}→{6,3} crossover, are stated without error bars or a discussion of realization-to-realization variation. Since the paper explicitly contrasts robust boundary observables with sensitive bulk DOS, the statistical evidence for the bulk sensitivity should be quantified; otherwise the contrast may partly reflect different finite-size and averaging procedures.
minor comments (5)
  1. [Throughout] The notation ⟨ρ⟩ is introduced but not defined precisely before Eq. (6); define it as the average over the M realizations of the realized fraction ρ̂_k. Also, the symbol ρ is used both for the target fraction and for the realized average; please clarify.
  2. [Fig. 4 captions] The fitted values A and b are shown in the figure panels, but no uncertainties are given. State the fit range and the number of data points used for each fit.
  3. [§III.B, Eq. (22)] Equation (22) is a nice simplification; it would help to state explicitly that for nearest-neighbor boundary sites A_ab contributes a contact term, which is why the power-law regime is probed at large d_ab.
  4. [§III.C, Eq. (28)] The definition of B as ‘all sites of the graph with coordination number 3’ is confusing because the construction in §II fixes the coordination number to 3 for every internal site; please clarify whether boundary sites with reduced coordination are excluded and how B differs between bulk and boundary in practice.
  5. [Appendix B, Table I] The standard deviations for N are given in parentheses, but the meaning (standard deviation across the 30 realizations) is not stated in the table caption; add a sentence.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the robustness claim is a numerical observation supported by fits, not a derivation equivalent to its inputs.

full rationale

The paper's central claims are empirical: it generates random defective {7,3}->{6,3} and {8,3}->{6,3} flakes, computes exact Gaussian boundary correlators and bulk DOS, and fits the correlators to exponential and power-law forms. The power-law in d is a mathematical consequence of the fitted exponential distance relation plus exponential decay in D (Eqs. 16-19), not an independent prediction, but the paper presents it as a characterization of the data and extracts xi and Delta with explicit fits (Fig. 5 and text). These fits could fail, and the paper reports deviations at large distances and for individual realizations, so the observation is not forced by construction. The formula for the boundary correlator (Eqs. 20-22) is a standard Gaussian/Schur-complement identity; the citation to Ref. [18] (which includes a co-author) is not load-bearing because the paper's novel content is the crossover behavior, not the pure-hyperbolic power law. Self-citations to [18] for the hyperbolic scaling form are benchmarked against data and against the known continuum relation m^2 l^2 = Delta(Delta-1). The acknowledged limitation that results are averaged over 30 realizations and individual flakes show larger deviations (Section III.B) is a robustness/correctness caveat, not circularity. Overall, no prediction reduces by definition to an input or to a self-citation chain.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. The main modeling inputs are the random-defect ensemble parameter rho, the fitted distance-scaling constants, and the fitted xi and Delta. Several physical identifications come from prior hyperbolic-lattice literature, including two papers with overlapping authorship.

free parameters (4)
  • hexagon fraction rho = 0.0-1.0 in steps such as 0.10(1), 0.90(1)
    Interpolation control parameter for the crossover. Realizations are post-selected with tolerance epsilon=0.01, so the realized fraction is close to rho. It enters all central claims.
  • distance fit parameters A, b = A ~ 2.19-5.48, b ~ 0.08-0.23 for {7,3}->{6,3} in Fig. 4
    Fitted to d_ab = A exp(b D_ab) for each rho. These fits encode the logarithmic distance compression that underlies the power-law boundary correlations.
  • correlation length xi and scaling dimension Delta = Values shown in Fig. 6 as functions of m^2 l^2
    Extracted by fitting the boundary two-point function to Eqs. (23) and (24). Their rho-dependence is the main evidence for boundary robustness.
  • DOS bin width deltaE = 0.1
    Regularization width in Eq. (27). The authors state that reasonable variations do not change qualitative results, so it is a minor free choice.
assumptions (5)
  • domain assumption Tile-by-tile inflation rules generate a planar degree-3 graph whose adjacency matrix encodes the chosen polygon tiling.
    The up/down word construction in Section II is assumed to correctly represent the planar graph of concentric polygon rings for arbitrary local p choices. All numerical results depend on this.
  • domain assumption The boundary two-point correlation function formula, Eq. (20)-(22), carries over from regular hyperbolic lattices to defective flakes.
    The formula is imported from Ref. [18] and reduced to A_ab + G_{i0(a)i0(b)}. Its validity for random hexagon/heptagon flakes is assumed, not re-derived.
  • domain assumption The tight-binding adjacency matrix approximates the Laplace-Beltrami operator on the Poincare disk with curvature radius l.
    The continuum identification in Eqs. (11)-(12) is taken from prior hyperbolic-lattice literature [3,18] and used to interpret masses and scaling dimensions.
  • domain assumption The holographic mass-dimension relation m^2 l^2 = Delta(Delta-1) remains a useful reference for defective lattices.
    The gray curve in Fig. 6 is the continuum AdS relation. The authors state it accurately describes hyperbolic data and gives a useful estimate for defective lattices, but this is an approximation for rho>0.
  • domain assumption The bulk DOS defined by selecting coordination-3 sites, Eq. (28), approximates the infinite-lattice DOS and removes boundary-mode contamination.
    This definition is imported from prior work [6]. The paper notes it is only approximate in hyperbolic limits and cannot be improved by larger flakes.

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Pith. "Pith review of Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices." pith.science (2026). https://pith.science/paper/JY2YLQ4U

@misc{pith2026260200231,
  author       = {Pith},
  title        = {Pith review of: Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JY2YLQ4U}},
  note         = {Machine review of arXiv:2602.00231}
}
read the original abstract

We study tight-binding models in the crossover from hyperbolic to Euclidean lattices, realized through the successive insertion of Euclidean defects into hyperbolic lattices. We analyze how the holographic two-point boundary correlation function and bulk density of states evolve as defects are gradually introduced. We find that bulk properties are strongly affected by the presence of Euclidean defects, whereas boundary observables remain remarkably robust even at high defect fractions. This robustness indicates that essential features of boundary physics on hyperbolic lattices, which capture aspects of anti de-Sitter/conformal field theory (AdS/CFT)-like dualities in discrete systems, can be reproduced both experimentally and numerically without requiring perfectly hyperbolic lattices, thereby reducing the system size needed for implementation.

Figures

Figures reproduced from arXiv: 2602.00231 by the authors.

Figure 1
Figure 1. FIG. 1. We show some realizations of the hyperbolic-to-Euclidean lattice crossover, where hexagonal faces are randomly inserted [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Construction of a defective [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: illustrates the relation between the different dis￾tance measures in Eqs. (15) and (16) in the hyperbolic-to￾Euclidean lattice crossover from {7, 3} → {6, 3}. It shows that the logarithmic relation between Dab and dab, char￾acteristic of hyperbolic lattices, remains va…
Figure 5
Figure 5. Figure 5: FIG. 5. Boundary two-point correlation functions [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Correlation length [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Bulk DOS in the hyperbolic-to-Euclidean crossover [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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Pith tools

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