{"id":"894f638d-8d31-4cbb-bf3a-2c5888d6a6de","arxiv_id":"2412.14722","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For uniform random graphs used as tight-binding lattices, the level spacing distribution switches from Wigner-Dyson to Poisson through semi-Poisson as the edge density decreases, claimed as a quantum phase transition.","lead":"This paper studies the energy levels of random graphs treated as quantum tight-binding models, and finds three patterns of level spacing (chaotic, localized, critical) as graph density changes. The authors call the change a phase transition, but the evidence is visual matching at two system sizes without thermodynamic scaling.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finite-size scaling gap is the load-bearing weakness: distributions at n=9000/18000 cannot distinguish a thermodynamic phase transition from a finite-size crossover without a quantitative order parameter and scaling analysis.","rationale":"The reader's weakest assumption—that the visually identified spacing distributions at two system sizes represent distinct thermodynamic phases rather than finite-size crossovers—is exactly the load-bearing concern. I agree that this assumption is unverified and that it is essential to the paper's central claim. The paper provides multiple qualitative supports (DOS approaching Wigner's semicircle, IPR increasing with decreasing R, level crossings becoming more frequent), but none of these is quantitative enough to distinguish a phase transition from a crossover. The manuscript itself flags no limitation on this point; it simply asserts scale invariance from two sizes. The absence of error bars, the lack of a quantitative order parameter, and the absence of any scaling collapse are concrete omissions, not stylistic choices. The hand-selection of Rc values further weakens the claim, as no objective criterion is given. A revision with full finite-size scaling, error bars, and a quantitative order parameter could potentially rescue the claim, and the underlying numerical observations may well be correct—but the current evidence does not support the stated conclusion. Hence REJECT with moderate confidence is appropriate, matching the reader's verdict.","tokens_in":6657,"tokens_out":1540,"duration_ms":11969,"concrete_test":"Perform a finite-size scaling analysis using the average adjacent gap ratio <r> (or IPR) for n=2000, 4000, 8000, 16000, 32000 at fixed R values near the purported critical points, with at least 1000 realizations and bootstrap error bars. If <r> versus R curves for different n collapse onto a single crossing point Rc(n) that converges to a finite Rc as n→∞ with consistent critical exponent nu, the phase-transition claim is supported; if the curves merely shift monotonically with n or the crossing moves systematically toward R=0.5, the observations are a finite-size crossover and the central claim fails.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim is that uniform random graphs G(n,m) undergo a genuine quantum phase transition, with Wigner-Dyson, semi-Poisson, and Poisson spacing distributions marking distinct metallic, critical, and localized phases. The evidence is P(S) curves at n=9000 and n=18000 for hand-selected R values (R≈1, 0.82, 0.7, 0.55), matched visually to the three analytic curves. This cannot establish a thermodynamic phase transition, because the same visual pattern is the expected finite-size crossover in a single phase: in Anderson-type systems, P(S) interpolates from Wigner to Poisson as system size increases relative to localization length. The paper asserts scale invariance only for the Wigner case at R=1 (two sizes), with no scaling collapse, no n-dependence study, no error bars, and no quantitative order parameter such as the integrated level-spacing ratio or the IPR scaling exponent. The IPR data in Fig. 4 are presented without error bars and without finite-size scaling, although the IPR at fixed n actually shows the expected crossover-like onset of localization. The hand-picked Rc values (e.g., Rc=0.66 for 0.5<E<0.6) are not derived from any criterion. Thus the strongest claim of distinct phases and universal critical behavior is unsupported; the weaker claim of a size-dependent crossover is plausible but not established with the statistics shown.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the energy-level statistics of uniform random graphs G(n,m) treated as tight-binding Hamiltonians. For several edge-to-vertex ratios R and graph sizes n=9000 and n=18000, the authors compute the density of states, the unfolded level-spacing distribution P(S), level crossings, and the inverse participation ratio. They report Wigner-Dyson statistics for dense graphs, Poisson statistics near R≈0.5, and semi-Poisson statistics at intermediate ratios, and they interpret these regimes as metallic, localized, and critical phases of a quantum phase transition whose critical ratio depends on energy.","tokens_in":6849,"tokens_out":4714,"duration_ms":42237,"significance":"If established, the result would connect random-graph geometry with quantum-chaos universality and would be of interest to both the quantum-chaos and network-science communities. The paper has clear strengths: the model is simple and well defined, the P(S) data are obtained by direct numerical unfolding over 1000 configurations rather than by fitting to the claimed analytic curves, and the IPR provides an independent look at wavefunction localization. However, the paper's central claim of a genuine thermodynamic phase transition is not supported by the presented evidence; the data are consistent with a finite-size crossover. The missing finite-size scaling and order-parameter analysis are in principle obtainable, so I regard the paper as requiring major revision rather than being irreparable.","major_comments":[{"comment":"The central claim of a phase transition rests entirely on the visual agreement of P(S) with the Wigner, Poisson, and semi-Poisson curves at n=9000 and n=18000. No goodness-of-fit test, no error bars, and no finite-size scaling or scaling collapse are reported. The statement that 'the Wigner distribution remains invariant under scaling as shown from the two cases n=9000 and n=18000' is not a demonstration of scale invariance: agreement at two system sizes is also the expected behavior of a finite-size crossover in a model with a large but finite localization length. The same data could be obtained in a single phase, so the phase-transition interpretation is not established.","section":"Fig. 2 and surrounding text"},{"comment":"The identification of the critical ratio Rc=0.66 for the window 0.5<E<0.6 is not based on any stated criterion. The text says that 'a phase transition ... at the critical ratio Rc=0.66' is found because the semi-Poisson distribution appears, but 'appears' is not defined. The values of Rc and the energy windows are therefore free parameters of the analysis, and the claim that Rc depends on energy is not quantitative. A defined order parameter with a crossing criterion, e.g., the integrated level-spacing ratio or the scaling of the IPR, is needed.","section":"Discussion after Eq. (4), specifically Rc=0.66"},{"comment":"The IPR data are presented for a single system size (n=9000) without error bars and without finite-size scaling. The text itself characterizes the behavior as an 'onset to localization' as R decreases, which is a crossover statement rather than evidence for distinct thermodynamic phases. In particular, the IPR curves in Fig. 4 do not distinguish localized wavefunctions in the thermodynamic limit from finite-size localization precursors, so they cannot support the phase-transition claim.","section":"Fig. 4 and IPR discussion"},{"comment":"The explanation that the transition is driven by the emergent spatial dimension D relies on average values ⟨D⟩=3.15, 2.64, and 1.90 for three values of R, but the method for computing D is not described in this manuscript and no direct quantitative relation between D and the observed P(S) or IPR is established. This is a speculative interpretive step, not evidence for the phase transition, and it should be presented as such or omitted from the main claims.","section":"Emergent-dimension explanation near the conclusion"}],"minor_comments":[{"comment":"The title contains a typo: 'rando m graphs' should be 'random graphs'. There are also several typographical errors in the text, including 'Furhter', 'wavefucntion', and 'occuring', which should be corrected.","section":"Title and text typos"},{"comment":"In Fig. 2d the panel includes a curve for n=18000-R=1 alongside the R=0.66 and R=0.55 curves, but the role of this reference curve is not explained in the caption or text.","section":"Fig. 2d"},{"comment":"The vertical axis of Fig. 3 appears to lack tick labels in the printed figure, and the caption does not state how many levels are shown or how the unfolding was performed for this plot.","section":"Fig. 3"},{"comment":"The emergent-dimension scaling method is cited to the authors' own previous work but is not summarized; a brief description of the method would make the paper more self-contained.","section":"References [8,9]"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript is very short and the gap between the numerical data and the phase-transition claim is large. I recommend major revision rather than rejection because the missing finite-size scaling and order-parameter analysis are, in principle, obtainable with additional numerical work. The direct comparison of P(S) to analytic benchmarks is not circular, but the interpretation in terms of thermodynamic phases and the claim of energy-dependent critical ratios need a much more quantitative basis. If the additional analysis does not produce a scaling collapse or a clear order-parameter signature, the paper should be rejected in a later round."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper documents a genuine numerical pattern — Wigner-Dyson to Poisson level spacing in uniform random graphs as the edge ratio R drops, with semi-Poisson statistics in between, and a critical R that shifts with energy. That is worth knowing. But the headline claim of a quantum phase transition between distinct thermodynamic phases is not established by the evidence. Two system sizes and visual agreement with analytic curves cannot separate a true transition from a finite-size crossover.\n\nWhat is actually new: the energy-dependent critical ratio and the explicit Wigner-Poisson-semi-Poisson sequence in G(n,m) graphs. Earlier work, including the authors' own, studied level statistics of various random graphs, but I don't think this precise observation is in the literature. The paper is clearly written, the unfolding procedure is standard, and the DOS panel showing the approach to the Wigner semicircle is a nice check. The IPR data are consistent with localization, and the connection to the emergent dimension from their prior papers is a reasonable interpretive step, not a fabricated one.\n\nThe soft spots are real and load-bearing. There is no finite-size scaling, no scaling collapse, no quantitative order parameter — the integrated level-spacing ratio is not computed — and no error bars on the P(S) curves or the IPR. The claim that the Wigner distribution at R=1 is scale invariant rests on two sizes, which proves little. The critical ratios (e.g., Rc=0.66 for 0.5<E<0.6) are picked by eye, with no criterion stated. Because the paper repeatedly calls these regimes 'phases' and a 'phase transition,' the missing scaling analysis is not a technical omission; it is the difference between a crossover and a phase transition. As written, the stronger claim is unsupported, though the weaker claim of a gradual crossover as R drops is entirely plausible.\n\nWho should read this: people working on quantum chaos on networks, sparse random matrices, and toy models of fluctuating-dimensional systems. The observation is suggestive and could be a useful starting point. Deserves a serious referee: yes, if the venue is willing to push for a proper scaling analysis. My recommendation is to send it to review, with the expectation that the authors add finite-size scaling, a quantitative order parameter such as the average spacing ratio, and error bars. Right now it is a solid short paper that overstates its conclusion.","headline":"Plausible numerical crossover, but the phase-transition claim outruns the data: no scaling analysis, no error bars, and the critical ratios are hand-picked.","tokens_in":7444,"tokens_out":1961,"would_cite":false,"duration_ms":15821,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Uniform random graphs, treated as quantum tight-binding lattices, undergo a phase transition from chaotic, extended states to localized, integrable states as the edge-to-vertex ratio is lowered, with a semi-Poisson critical regime in…","keywords":["level spacing statistics","random graphs","Anderson localization","quantum chaos","Wigner-Dyson distribution","semi-Poisson statistics","inverse participation ratio","tight-binding model"],"falsifier":"A decisive check is to measure the spacing distribution and the inverse participation ratio scaling at a fixed critical ratio, say R=0.82 in the window 0.1<E<0.2, as the graph size doubles to n=36000 and n=72000: true criticality requires the semi-Poisson form to persist and the inverse participation ratio to scale with a nontrivial fractal exponent, while drift toward the Wigner or Poisson form (or IPR scaling like 1/n) would falsify the phase-transition interpretation.","tokens_in":6343,"feed_emoji":"⚛️","tokens_out":8945,"duration_ms":67807,"temperature":0.7,"pith_summary":"The paper argues that a uniform random graph G(n,m), viewed as a lattice for a quantum particle, is a self-contained disordered system whose spectral statistics change universality class as the ratio of edges to vertices R is varied. For dense graphs (large R), level spacings follow the Wigner-Dyson distribution, indicating a metallic, quantum-chaotic phase; for sparse graphs near R=0.5, they follow the Poisson distribution, indicating localized, integrable behavior. In between, the paper finds a semi-Poisson distribution, which it identifies as the critical regime of a quantum phase transition, with the critical value of R depending on energy. If this is right, random geometry alone—without any on-site disorder—can drive Anderson-type localization transitions, and systems with fluctuating spatial dimension can exhibit phase-transition universality.","feed_headline":"Random graphs show a chaos-to-localization phase transition","feed_subtitle":"Level-spacing statistics shift from Wigner-Dyson to Poisson as the edge-to-vertex ratio falls, with semi-Poisson at criticality.","key_machinery":"The central object is the uniform random graph $G(n,m)$ with $m$ edges on $n$ vertices, whose adjacency matrix serves directly as the tight-binding Hamiltonian. The control parameter is the ratio $R=m/n$, which acts as the disorder strength: it sets how many hopping paths a wavefunction can take and, through the giant-component transition at $R=0.5$, determines whether a macroscopic connected lattice exists at all. The diagnostic is the unfolded level-spacing distribution $P(S)$, compared against the Wigner-Dyson, Poisson, and semi-Poisson forms, supplemented by the inverse participation ratio for the wavefunctions. The argument's load-bearing link is the relation between $R$ and the emergent spatial dimension $D$, computed by the scaling method of the authors' earlier work, so that the metallic-to-localized transition is recast as a dimensional crossover from $D\\approx 3$ to $D\\approx 2$.","core_discovery":"The central discovery is that the level-spacing statistics of uniform random graphs realize the three classic regimes of the Anderson transition, controlled not by a random potential but by the graph's edge-to-vertex ratio $R=m/n$. For a fixed energy window near the band center, dense graphs ($R\\approx 1$–$2$) yield Wigner-Dyson spacing $P(s)=(\\pi/2)s\\,e^{-\\pi s^2/4}$, the signature of chaotic, extended wavefunctions; sparse graphs near the structural transition $R=0.5$ yield Poisson spacing $P(s)=e^{-s}$, the signature of localized, integrable wavefunctions; intermediate ratios (e.g., $R\\approx 0.8$ at $0.1<E<0.2$) yield semi-Poisson $P(s)=4s\\,e^{-2s}$, which the paper reads as the critical point of a phase transition. The critical ratio depends on energy: in the window $0.5<E<0.6$ the semi-Poisson distribution appears at $R_c\\approx 0.66$. The inverse participation ratio confirms that wavefunctions become more localized as $R$ decreases, and the authors connect the transition to a drop in the emergent spatial dimension from about 3.15 at $R=1$ to 1.90 at $R=0.6$.","pith_inferences":["A testable consequence beyond the paper: random regular graphs with fixed degree should show the same Wigner-to-Poisson transition as the degree is lowered, which would show that connectivity density, not degree fluctuations, drives the transition.","One step further, one could measure the multifractal spectrum of the wavefunctions at the claimed critical ratios; genuine criticality would give a nontrivial spectrum of inverse participation ratio exponents, and a trivial one would instead indicate a finite-size crossover.","The energy dependence of the critical ratio suggests that the full spectral phase diagram in the $(E,R)$ plane could have a mobility-edge structure worth mapping explicitly, a prediction the paper leaves implicit.","If the semi-Poisson regime is truly critical, it should appear at a well-defined critical value of $R$ in the thermodynamic limit; testing the drift of $R_c$ with $n$ would separate a true transition from a crossover."],"forward_implications":["If the transition is a genuine phase transition, uniform random graphs provide a minimal model in which disorder is purely geometric: no on-site potentials or magnetic fields are needed to produce Anderson-like localization.","The energy-dependent critical ratio $R_c(E)$ implies a mobility edge in the $(E,R)$ plane, so a complete phase diagram could be mapped by spectral window.","The appearance of semi-Poisson statistics identifies the critical regime with the universality class of the conventional Anderson transition, suggesting common critical behavior despite the absence of a fixed spatial dimension.","The dimensional-crossing interpretation predicts that other random geometries with tunable connectivity will show analogous Wigner-to-Poisson transitions.","Wavefunction localization in graphs could be directly probed via transport or conductance calculations, connecting spectral statistics to physical observables."],"supporting_citations":[{"why":"It supplies the G(n,m) random graph model and the structural phase transition at R=0.5 where a giant component emerges, anchoring the localization threshold.","marker":"[5]"},{"why":"It provides the scaling method used to compute the emergent spatial dimension D of the graph, linking R to an effective dimensionality.","marker":"[8]"},{"why":"It is the companion preprint with the scaling approach for emergent dimension that the paper uses to interpret the transition as a dimensional crossover.","marker":"[9]"},{"why":"It establishes the level-spacing distribution as a probe of localization in disordered lattices, the diagnostic the paper applies to graphs.","marker":"[10]"},{"why":"It supplies the reference Anderson transition in three dimensions where Wigner-Dyson, semi-Poisson, and Poisson statistics appear across the critical disorder.","marker":"[11]"},{"why":"It supports the identification of semi-Poisson statistics as the critical distribution between metallic and localized phases.","marker":"[12]"},{"why":"It documents chaotic Wigner-Dyson level statistics on other graph-like systems, providing precedent for the metallic phase.","marker":"[24]"}],"fun_headline_variants":["Random graph geometry drives chaos-to-localization transition","Edge-to-vertex ratio tunes quantum chaos in random graphs","Wigner-Dyson to Poisson: random graphs hold Anderson transition","Random graph connectivity dictates quantum phases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the visually identified Wigner, Poisson, and semi-Poisson spacing distributions at only two graph sizes (n=9000 and n=18000) represent distinct thermodynamic phases in the infinite-size limit, rather than a smooth finite-size crossover of a single phase.","fun_headline_variants_meta":{"raw":{"variants":["Random graph geometry drives chaos-to-localization transition","Edge-to-vertex ratio tunes quantum chaos in random graphs","Wigner-Dyson to Poisson: random graphs hold Anderson transition","Random graph connectivity dictates quantum phases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000665,"raw_usage":{"total_tokens":3092,"prompt_tokens":1058,"completion_tokens":2034,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":1973}},"tokens_in":674,"tokens_out":2034,"duration_ms":12612,"temperature":1.0,"reasoning_tokens":1973,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:57:51.561680+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check is to measure the spacing distribution and the inverse participation ratio scaling at a fixed critical ratio, say R=0.82 in the window 0.1<E<0.2, as the graph size doubles to n=36000 and n=72000: true criticality requires the semi-Poisson form to persist and the inverse participation ratio to scale with a nontrivial fractal exponent, while drift toward the Wigner or Poisson form (or IPR scaling like 1/n) would falsify the phase-transition interpretation.","supporting_citations":[{"cited_title":"Frieze, M","cited_arxiv_id":null,"evidence_quote":"It supplies the G(n,m) random graph model and the structural phase transition at R=0.5 where a giant component emerges, anchoring the localization threshold."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It establishes the level-spacing distribution as a probe of localization in disordered lattices, the diagnostic the paper applies to graphs."},{"cited_title":"Evangelou, Phys","cited_arxiv_id":null,"evidence_quote":"It supplies the reference Anderson transition in three dimensions where Wigner-Dyson, semi-Poisson, and Poisson statistics appear across the critical disorder."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supports the identification of semi-Poisson statistics as the critical distribution between metallic and localized phases."},{"cited_title":"Smilansky, Chaos","cited_arxiv_id":null,"evidence_quote":"It documents chaotic Wigner-Dyson level statistics on other graph-like systems, providing precedent for the metallic phase."}],"review_version":1}