{"id":"bcec8e90-ed68-4345-a094-4e0b18aaa795","arxiv_id":"1908.09469","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":11,"one_line_summary":"Simulations of random planar maps and discrete Liouville quantum gravity contradict Watabiki's formula for the Hausdorff dimension and support the Ding-Gwynne formula for central charges in [-12.5, 0).","lead":"This paper uses computer simulations of random surfaces to measure their fractal dimension, and finds that a 25-year-old formula is wrong. The new data favor an alternative formula, which is important for understanding quantum gravity in two dimensions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The c<-12.5 portion of the claim relies on DLFPP small-xi scaling that the paper itself calls unreliable; Table 6's errors do not cover ansatz failure.","rationale":"The planar-map data for gamma = sqrt(8/3), sqrt(2), sqrt(4/3), 1 are the strongest part of the paper: the scaling collapses are good, two distance measures agree, the fitted dimensions are many sigma away from Watabiki in the c in [-12.5,0) range, and the code and data are released. That part of the central claim is robust. The load-bearing weak point is the extension to smaller gamma via DLFPP: the authors themselves state that for xi<0.35 the scaling is markedly worse, that the systematic errors are larger, and that they are hesitant to rule out lambda_DG = xi/sqrt(6). Since this small-xi, c<-12.5 regime is the only evidence for the quantitative negative deviation from Ding-Gwynne and for the abstract's claim about all c<0, the paper should either present those measurements with a clearly restricted scope or add larger-lattice tests before the claim is taken as settled. The reader already issued a CONDITIONAL verdict, and this concern is essentially the same weakest assumption, so no further verdict adjustment is needed.","tokens_in":17850,"tokens_out":12531,"duration_ms":136087,"concrete_test":"Run the released DLFPP code at w=2^15 and w=2^16 for xi=0.05, 0.1, 0.15, and 0.2, using the same reference-distribution collapse and the nu,s variation of Section 5, and compare the lambda estimates from ansatz (38) with Table 6. If lambda drifts toward xi/sqrt(6) with increasing w, the small-xi deficit is a finite-size artifact and the c<-12.5 part of the claim should be downgraded; if it stabilizes below xi/sqrt(6), the deficit is real. An independent cross-check would be the disk-covering Liouville graph distance of Section 4.1 for one or two small gamma values.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is two-part: (i) Watabiki's formula fails for every simulated c<0, and (ii) Ding-Gwynne holds for the most reliable range c in [-12.5,0). Part (ii) is supported by the four planar-map models, whose collapses are good and whose estimates are far from Watabiki at the quoted precision. Part (i) for c<-12.5, however, rests entirely on the DLFPP measurements in Table 6 for xi<~0.25, i.e. gamma<~0.5 and c<~-100. The paper states in Section 6 that for xi<0.35 the finite-size scaling is markedly worse and that the authors are hesitant to rule out lambda_DG = xi/sqrt(6) on the basis of current data. The systematic errors in Table 6 are obtained by varying the fit window nu and shift s inside a single assumed ansatz, equations (37)-(38); they do not test whether that ansatz is correct. Figure 13 shows visibly worse collapse at small xi, and the long straight lattice segments mentioned in Section 6 imply near-Euclidean corrections with a different exponent that could persist to much larger w. Thus the quantitative lambda and d_gamma values at c<-12.5, and the claimed negative deviation from Ding-Gwynne, are not securely established. The abstract's blanket statement of contradiction with Watabiki for all simulated c<0 should be restricted to the range c in [-12.5,0).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents new numerical estimates of the Hausdorff dimension $d_\\gamma$ of two-dimensional Liouville quantum gravity for central charges $c<0$, using two complementary approaches: finite-size scaling of graph and dual graph distances in four exactly samplable models of random planar maps (uniform, spanning-tree-decorated, bipolar-oriented, and Schnyder-wood-decorated), and discrete Liouville first passage percolation (DLFPP) on the torus for $\\xi\\in[0.01,0.4]$. The planar-map data give estimates that lie close to the Ding--Gwynne formula $d_{\\rm DG}=2+\\gamma^2/2+\\gamma/\\sqrt{6}$ and many standard deviations away from Watabiki's formula for $\\gamma=\\sqrt{2},\\sqrt{4/3},1$. The DLFPP data also favor $\\lambda_{\\rm DG}=\\xi/\\sqrt{6}$ for $\\xi\\gtrsim0.35$, with increasingly visible negative deviations at smaller $\\xi$. The paper concludes that Watabiki's formula is contradicted for all simulated $c<0$, while the most reliable data in $c\\in[-12.5,0)$ agree with the Ding--Gwynne formula.","tokens_in":18338,"tokens_out":7410,"duration_ms":73872,"significance":"If the planar-map estimates withstand scrutiny, this is an important result: it provides high-precision numerical evidence against a twenty-five-year-old empirical formula and supports a recent rigorous-motivated alternative. The paper's strengths include the use of exactly samplable decorated planar maps, the independent graph and dual graph distance observables, the very small statistical errors, and the public release of source code and data. The main weakness is the reliance on an assumed finite-size scaling ansatz with fitted shift and correction parameters, and the small-$\\xi$ DLFPP regime, which carries the full weight of the $c<-12.5$ part of the claim, is explicitly acknowledged to have poor scaling. The core planar-map results are likely to be a lasting contribution even if the small-$\\xi$ interpretation is later revised.","major_comments":[{"comment":"The statement that the estimates are in clear contradiction with Watabiki's formula for all simulated values of $c\\in(-\\infty,0)$ is too strong for the range $c<-12.5$. In that regime the estimates come entirely from the DLFPP analysis in Table 6 with $\\xi\\lesssim0.25$, and Section 6 explicitly states that the finite-size scaling is markedly worse for small $\\xi$ and that the authors are hesitant to rule out $\\lambda_{\\rm DG}=\\xi/\\sqrt{6}$ on the basis of current data. The systematic errors in Table 6 are obtained by varying the fit parameters $\\nu$ and $s$ inside the fixed ansatz (37)--(38); they do not test whether that ansatz is correctly specified. The abstract should restrict the contradiction claim to the range $c\\in[-12.5,0)$ supported by the planar-map data, and describe the $c<-12.5$ results as provisional or suggestive rather than conclusive.","section":"Abstract and Section 6"},{"comment":"The quoted errors on the planar-map estimates of $d_\\gamma$ are statistical errors from fits to the assumed form (22), with the shift $s$ fixed by the data-dependent procedure described in Section 3. No systematic error is reported for the choice of $s$, for the correction exponent $\\delta$, or for the correction form itself. This matters because the claimed agreement with Ding--Gwynne is at the $10^{-3}$ level (for example, $d=3.5791\\pm0.0033$ versus $d_{\\rm DG}=3.5774$ for model S), and a modest change in the shift or correction term could shift the fitted $d$ by more than the quoted statistical error. A robustness analysis should be added, for example varying $s$ over the values that still produce visually acceptable collapses, omitting the shift, or using an alternative correction functional form, and the discussion should state whether the separation from Watabiki remains many sigma under those variations.","section":"Section 3 and Tables 4-5"},{"comment":"The conversion from the measured exponent $\\lambda$ to $d_\\gamma$ in Table 6 relies on the heuristic relation $\\xi=\\gamma/d_\\gamma$ in Eq. (28) together with the scaling relation (31). For $\\xi\\lesssim0.25$, the negative deviation from $\\lambda_{\\rm DG}$ is therefore interpretable as a deviation in $d_\\gamma$ only if both (28) and the pointwise convergence assumption (37) are correct. The paper should state this double dependence explicitly and should test the small-$\\xi$ results against an alternative analysis, for example by attempting to model the Euclidean straight-segment contribution to DLFPP geodesics mentioned in Section 6, whose scaling exponent is close to $1-\\lambda$ for small $\\xi$ and could contaminate the finite-size estimate at the lattice sizes used.","section":"Section 5 and Table 6"}],"minor_comments":[{"comment":"The rows for dual graph distance are indicated only by a colon after the model letter (for example, '(U):'); please relabel these rows explicitly as 'U (dual)' to avoid confusion.","section":"Table 4"},{"comment":"The footnote after Eq. (19) reads 'a limit in distribution as $n\\to\\infty$ followed by an almost sure limit as $n\\to\\infty$'; the second limit should presumably be as $r\\to\\infty$, and the sentence should be corrected.","section":"Section 3, footnote"},{"comment":"The text does not explain how the error bars on the logarithmic ratios in Figure 9 are propagated from the fitted values of $k_n$; one sentence describing the error propagation would be helpful.","section":"Eq. (21) and Figure 9"},{"comment":"The central charge value for $\\xi=0.01$ is written as $-59.6k$; please use standard scientific notation such as $-5.96\\times10^4$.","section":"Table 6"},{"comment":"The phrase 'for all simulated values of $c\\in(-\\infty,0)$' is informal because only finitely many values are simulated; please write 'for all simulated $c<0$' or specify the actual simulated range.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper merits major revision rather than rejection: the planar-map evidence against Watabiki's formula is strong and would be a significant contribution, but the abstract's universal claim extends beyond what the DLFPP data support, and the small-$\\xi$ systematic errors are not yet adequate. I have no concerns about the fit with the journal's scope or about the novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Barkley and Budd have put together a serious piece of numerical work that effectively settles the question for c in [-12.5,0): Watabiki's formula is wrong, and the Ding-Gwynne formula fits very well. The four planar map estimates are precise, the two new models (bipolar-oriented triangulations and Schnyder-wood-decorated triangulations) are well chosen and can be sampled with the efficient lattice-walk bijections, and the DLFPP scan over xi in [0.01,0.4] is much broader than anything before. The source code and data are on Zenodo, so the numbers are reproducible.\n\nThe finite-size scaling collapses for the planar maps are genuinely good. For gamma = 1, sqrt(4/3), sqrt(2), sqrt(8/3), the estimates sit on top of d_DG and are many sigma away from d_W. I would not bet against that part of the paper.\n\nThe soft spots are in the small-gamma DLFPP regime. For xi below about 0.35, the collapse is visibly worse, and Section 6 says so explicitly. The systematic errors in Table 6 are obtained by varying the fit window and shift inside a single assumed scaling form; that does not cover a misspecified ansatz. The negative deviation from Ding-Gwynne around xi ~ 0.1 is visible in the plots, but the authors themselves are hesitant to rule out lambda_DG on that data. The abstract, however, says 'clear contradiction with Watabiki's formula for all simulated values of c in (-inf,0).' That is too strong. The reliable contradiction is in the window c in [-12.5,0). For more negative c, the data are suggestive but not conclusive.\n\nA minor issue: the planar map error bars do not include a systematic scan over the fitted shift s and the fit threshold. The consistency between graph and dual distance estimates makes me think this would not move the numbers much, but it is worth reporting.\n\nThis is a good paper, honestly written, with reproducible data. The central claim holds up in the regime where the authors trust their scaling. I would send it to a serious referee, and the referee should ask them to soften the c < -12.5 statements and add the systematic checks for the planar map fits. Readers in LQG and random planar maps will want to see this.","headline":"Strong numerical evidence against Watabiki's formula for the Hausdorff dimension in 2D quantum gravity, but the abstract overreaches for the small-gamma DLFPP regime.","tokens_in":18827,"tokens_out":3383,"would_cite":true,"duration_ms":30856,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C45","60D05","81T40"],"pacs":[],"model":"deepseek-v4-flash","headline":"High-precision simulations of random planar maps and Liouville first-passage percolation rule out Watabiki's formula for the Hausdorff dimension of two-dimensional quantum gravity, favoring the Ding-Gwynne formula for $c\\in[-12.5,0)$.","keywords":["Hausdorff dimension","two-dimensional quantum gravity","Liouville quantum gravity","random planar maps","Liouville first passage percolation","finite-size scaling","Watabiki formula","Ding-Gwynne formula"],"falsifier":"Run DLFPP at $\\xi=0.1$ on lattices $w=2^{13}$ through $2^{16}$ with the same shift-and-correction fitting protocol: Watabiki predicts $\\lambda=\\xi^2=0.01$, Ding-Gwynne predicts $\\lambda=\\xi/\\sqrt{6}\\approx0.0408$, and the paper's current estimate is $0.0341\\pm0.0006$. If the fitted $\\lambda(w)$ moves toward $0.0408$ as $w$ grows, the paper's support for Ding-Gwynne is confirmed; if it levels off near $0.0341$ or falls toward $0.01$, the small-$\\gamma$ conclusion collapses.","tokens_in":17648,"feed_emoji":"📐","tokens_out":15286,"duration_ms":132663,"temperature":0.7,"pith_summary":"Two-dimensional quantum gravity produces random fractal geometries whose Hausdorff dimension should depend on the central charge $c$ of the coupled matter system, and for decades Watabiki's formula was the only prediction consistent with simulations. This paper presents new finite-size scaling estimates from two independent numerical routes---four special random-planar-map universality classes and discrete Liouville first passage percolation on a torus---and reports that the data contradict Watabiki's formula for every simulated $c<0$. For $c\\in[-12.5,0)$ the most reliable estimates agree with the alternative Ding-Gwynne formula $d_\\gamma=2+\\gamma^2/2+\\gamma/\\sqrt{6}$. For $c<-12.5$ the estimates fall below that formula, but the scaling is visibly less accurate there, so the paper treats that regime as not decisive.","feed_headline":"2D gravity's fractal dimension contradicts Watabiki formula","feed_subtitle":"High-precision map and lattice simulations favor the Ding-Gwynne prediction for central charges below zero.","key_machinery":"The workhorse is the finite-size scaling ansatz (19)--(22): for a model of size $n$, the rescaled distance density $n^{1/d}\\rho_n(n^{1/d}x)$ is assumed to converge pointwise to a universal limit, with a constant shift $s$ and a power-law correction exponent $\\delta$ absorbing the leading finite-size effects. Fitting the scaling factors $k_n$ to this ansatz produces the exponent $d$ from $k_n\\sim (n/n_0)^{-1/d}$. The four planar-map models are sampled efficiently through bijections to walks in the quadrant, allowing ensembles with up to $2^{24}$ faces; the Liouville quantum gravity side uses discrete first passage percolation through the discrete Gaussian free field on $w\\times w$ tori, with the same collapse ansatz yielding $\\lambda(\\xi)$, which is converted to $d_\\gamma$ via $\\xi=\\gamma/d_\\gamma$ and $\\lambda=1-\\xi Q$.","core_discovery":"The central claim is that the Hausdorff dimension $d_\\gamma$ of two-dimensional quantum gravity is not the Watabiki prediction $d^W_\\gamma=1+\\gamma^2/4+\\sqrt{(1+\\gamma^2/4)^2+\\gamma^2}$ for the range $\\gamma\\in(0,2]$ that corresponds to $c<0$. In four random-planar-map models with $\\gamma=\\sqrt{8/3}$, $\\sqrt{2}$, $\\sqrt{4/3}$, and $1$, finite-size scaling of graph and dual-graph distances gives $d_{\\sqrt{8/3}}=3.9970\\pm0.0013$, $d_{\\sqrt{2}}=3.5791\\pm0.0033$, $d_{\\sqrt{4/3}}=3.1375\\pm0.0017$, and $d_1=2.9074\\pm0.0009$. These numbers agree with $d^{DG}_\\gamma=2+\\gamma^2/2+\\gamma/\\sqrt{6}$ to within about $0.001$--$0.003$ in each case, whereas Watabiki's formula misses the last two models by $0.04$--$0.06$. The Liouville-quantum-gravity side, measured by discrete first passage percolation with $\\xi\\in[0.01,0.4]$ and converted via $\\xi=\\gamma/d_\\gamma$ and $\\lambda=1-\\xi Q$ with $Q=2/\\gamma+\\gamma/2$, agrees with the Ding-Gwynne prediction $\\lambda^{DG}=\\xi/\\sqrt{6}$ for $\\xi=0.35,0.375,0.4$; at smaller $\\xi$ the estimates deviate negatively, and the paper attributes this to markedly worse finite-size scaling rather than to a settled failure of the formula.","pith_inferences":["If the small-$\\xi$ negative deviation is real rather than a scaling artifact, the true $d_\\gamma$ must cross from below Ding-Gwynne at small $\\gamma$ to agreement near $\\gamma=\\sqrt{2}$, forcing an inflection near $\\gamma\\approx0.2$; such a shape is not present in either simple formula.","The same collapse analysis applied to mated-CRT maps, which the paper names as a candidate for arbitrary $\\gamma$, would provide a third independent route and would avoid the straight-segment contamination that limits DLFPP at small $\\xi$.","A sharper test of the paper's methodology is that graph and dual-graph distance estimates approach the same $d_\\gamma$ from opposite sides; pushing both to larger map sizes should make the two correction signs converge, and any persistent gap would indicate the shift-correction ansatz is incomplete."],"forward_implications":["Watabiki's formula is numerically excluded across the whole simulated range $c<0$; for instance the $\\gamma=1$ planar-map estimate $2.9074\\pm0.0009$ is many standard deviations above Watabiki's $2.8508$ but within error of Ding-Gwynne's $2.9083$.","For $c\\in[-12.5,0)$ the combined planar-map and DLFPP data are consistent with $d_\\gamma=2+\\gamma^2/2+\\gamma/\\sqrt{6}$, making this the best current description of the fractal dimension in that range.","Because recent rigorous work has identified the Hausdorff dimension of Liouville quantum gravity with the scaling limits of these planar-map universality classes, the measured exponents transfer across all equivalent models in the same class.","For $c<-12.5$ the negative deviation from Ding-Gwynne means the exact asymptotic formula at small $\\gamma$ remains open; the rigorous lower bound $d_\\gamma\\ge 2+C\\gamma^{4/3}/\\log(\\gamma^{-1})$ is compatible with either behavior, so this regime is the natural next target."],"supporting_citations":[{"why":"supplies Watabiki's formula (1), the prediction the paper sets out to test against new data.","marker":"[16]"},{"why":"provides the earlier spanning-tree triangulation estimate $d_{\\sqrt{2}}\\approx3.58$ that was the main numerical support for Watabiki's formula.","marker":"[17]"},{"why":"gives the $c=-5$ Ising-coupled gravity estimate listed in Table 1 as previous data consistent with Watabiki.","marker":"[19]"},{"why":"gives the $c=-20$ semi-classical estimate listed in Table 1, one of the best earlier data points near the Watabiki prediction.","marker":"[21]"},{"why":"provides the high-precision toroidal estimates for $\\gamma=\\sqrt{2},\\sqrt{3},\\sqrt{10/3}$ that the new planar-map measurements are compared with.","marker":"[22]"},{"why":"supplies the earlier Liouville-quantum-gravity numerical measurement drawn in Figure 2b, the direct predecessor of the DLFPP estimates.","marker":"[23]"},{"why":"proves the rigorous lower bound (2) showing Watabiki's formula cannot hold as $\\gamma$ tends to 0, motivating the contradiction.","marker":"[24]"},{"why":"establishes universality and monotonicity of the LQG Hausdorff dimension, proposes the Ding-Gwynne formula (5), and supplies the bounds used in Figure 2a.","marker":"[25]"},{"why":"proves that discrete Liouville first passage percolation approximates the continuum distance, justifying the scaling relation (36) used to extract $\\lambda(\\xi)$.","marker":"[29]"},{"why":"gives the explicit geodesic-distance limit for uniform quadrangulations that supports the pointwise finite-size scaling ansatz (19).","marker":"[47]"}],"fun_headline_variants":["2D gravity fractal dimension overturns Watabiki formula","Precision simulations favor Ding-Gwynne over Watabiki","Hausdorff dimension of 2D gravity: Watabiki wrong, Ding-Gwynne right","Watabiki formula falsified by 2D gravity simulations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The extraction assumes that the finite-size scaling ansatz (19)--(22) holds for these non-uniform planar maps and for the small-$\\xi$ DLFPP data, namely that the rescaled distance density converges with a constant shift and a single power-law correction, and if that ansatz is misspecified, the fitted Hausdorff dimensions are biased.","fun_headline_variants_meta":{"raw":{"variants":["2D gravity fractal dimension overturns Watabiki formula","Precision simulations favor Ding-Gwynne over Watabiki","Hausdorff dimension of 2D gravity: Watabiki wrong, Ding-Gwynne right","Watabiki formula falsified by 2D gravity simulations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001037,"raw_usage":{"total_tokens":4484,"prompt_tokens":1187,"completion_tokens":3297,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":803,"completion_tokens_details":{"reasoning_tokens":3218}},"tokens_in":803,"tokens_out":3297,"duration_ms":23849,"temperature":1.0,"reasoning_tokens":3218,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:10:37.596373+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run DLFPP at $\\xi=0.1$ on lattices $w=2^{13}$ through $2^{16}$ with the same shift-and-correction fitting protocol: Watabiki predicts $\\lambda=\\xi^2=0.01$, Ding-Gwynne predicts $\\lambda=\\xi/\\sqrt{6}\\approx0.0408$, and the paper's current estimate is $0.0341\\pm0.0006$. If the fitted $\\lambda(w)$ moves toward $0.0408$ as $w$ grows, the paper's support for Ding-Gwynne is confirmed; if it levels off near $0.0341$ or falls toward $0.01$, the small-$\\gamma$ conclusion collapses.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies Watabiki's formula (1), the prediction the paper sets out to test against new data."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the earlier spanning-tree triangulation estimate $d_{\\sqrt{2}}\\approx3.58$ that was the main numerical support for Watabiki's formula."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the $c=-5$ Ising-coupled gravity estimate listed in Table 1 as previous data consistent with Watabiki."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the $c=-20$ semi-classical estimate listed in Table 1, one of the best earlier data points near the Watabiki prediction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the high-precision toroidal estimates for $\\gamma=\\sqrt{2},\\sqrt{3},\\sqrt{10/3}$ that the new planar-map measurements are compared with."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the earlier Liouville-quantum-gravity numerical measurement drawn in Figure 2b, the direct predecessor of the DLFPP estimates."},{"cited_title":"Upper bounds on Liouville first passage percolation and Watabiki's prediction","cited_arxiv_id":"1610.09998","evidence_quote":"proves the rigorous lower bound (2) showing Watabiki's formula cannot hold as $\\gamma$ tends to 0, motivating the contradiction."},{"cited_title":"The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds","cited_arxiv_id":"1807.01072","evidence_quote":"establishes universality and monotonicity of the LQG Hausdorff dimension, proposes the Ding-Gwynne formula (5), and supplies the bounds used in Figure 2a."},{"cited_title":"Comparison of discrete and continuum Liouville first passage percolation","cited_arxiv_id":"1904.09285","evidence_quote":"proves that discrete Liouville first passage percolation approximates the continuum distance, justifying the scaling relation (36) used to extract $\\lambda(\\xi)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the explicit geodesic-distance limit for uniform quadrangulations that supports the pointwise finite-size scaling ansatz (19)."}],"review_version":1}