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Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity

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Pith's one-line read 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 $c\in[-12.5,0)$.

desk verdict 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. read the letter →

arxiv 1908.09469 v1 pith:OIUJE4HF submitted 2019-08-26 gr-qc math-phmath.MP

Jerome Barkley , Timothy Budd This is my paper · ORCID
classification gr-qcmath-phmath.MP MSC 83C4560D0581T40
keywords Hausdorffdimensiontwo-dimensionalquantumgravityLiouvillerandomplanarmapsfirstpassagepercolationfinite-sizescalingWatabikiformulaDing-Gwynne
topics Quantum Gravity
open problems Quantum Gravity
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

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.

What carries the argument

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 $\lambda=1-\xi Q$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 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.

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 (3)
  1. [Abstract and Section 6] 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.
  2. [Section 3 and Tables 4-5] 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.
  3. [Section 5 and Table 6] 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.
minor comments (5)
  1. [Table 4] 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.
  2. [Section 3, footnote] 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.
  3. [Eq. (21) and Figure 9] 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.
  4. [Table 6] The central charge value for $\xi=0.01$ is written as $-59.6k$; please use standard scientific notation such as $-5.96\times10^4$.
  5. [Abstract] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the measured Hausdorff dimensions are produced independently and then compared with external formulas.

full rationale

The paper's central results are numerical estimates of the Hausdorff dimension obtained by finite-size scaling of graph distances in four random planar map models and of the Liouville first passage percolation exponent lambda from discrete Gaussian free field simulations. In both cases the target quantities are extracted from the data via scaling collapses and the fitted ansatze (19)-(22) and (37)-(38); neither Watabiki's formula (1) nor the Ding-Gwynne formula (5) is used as an input to these fits. The shift parameters and correction exponents are fitted to improve the collapse and quantify finite-size effects, but the predicted value of d_gamma is not a fitted parameter of either ansatz. The conversion from lambda to d_gamma in Section 5 uses the rigorous relation (31) together with the scaling heuristic (28); this is a coordinate change, not an import of the formula being tested. The planar-map estimates for gamma = 1, sqrt(4/3), sqrt(2), sqrt(8/3) are compared with d_W and d_DG only after they have been determined, so the agreement with d_DG is an external comparison rather than a construction. The DLFPP estimates for very small xi are explicitly flagged by the authors as having poor scaling and larger systematic uncertainty, and the authors state that they are hesitant to rule out lambda_DG on the basis of current data; this candor addresses correctness risk, not circularity. The self-citations to prior work by one of the authors ([21], [23]) provide earlier numerical estimates for comparison and are not load-bearing for the new determination; the load-bearing mathematical identifications of the models with Liouville quantum gravity are citations to independent results by other authors. Thus the derivation chain is self-contained with respect to the claims it tests, and no step reduces by construction to its own inputs.

Assumptions & free parameters 11 free parameters · 5 assumptions · 0 invented entities

The paper measures d_gamma by fitting data collapse, relying on scaling and universality assumptions from the literature and a phenomenological correction ansatz. The only numbers chosen by hand are the shift parameters and analysis thresholds; no new physical entities are introduced.

free parameters (11)
  • Shift parameter s for graph distance, uniform quadrangulations (U) = 0.940
    Fitted to optimize data collapse in Eq (19); affects extracted d.
  • Shift parameter s for dual graph distance, uniform quadrangulations (U) = 4.608
    Fitted to optimize data collapse in Eq (19); affects extracted d.
  • Shift parameter s for graph distance, spanning-tree-decorated quadrangulations (S) = 0.557
    Fitted to optimize data collapse in Eq (19); affects extracted d.
  • Shift parameter s for dual graph distance, spanning-tree-decorated quadrangulations (S) = 3.019
    Fitted to optimize data collapse in Eq (19); affects extracted d.
  • Shift parameter s for graph distance, bipolar-oriented triangulations (B) = 0.359
    Fitted to optimize data collapse in Eq (19); affects extracted d.
  • Shift parameter s for dual graph distance, bipolar-oriented triangulations (B) = 2.941
    Fitted to optimize data collapse in Eq (19); affects extracted d.
  • Shift parameter s for graph distance, Schnyder-wood-decorated triangulations (W) = 0.439
    Fitted to optimize data collapse in Eq (19); affects extracted d.
  • Shift parameter s for dual graph distance, Schnyder-wood-decorated triangulations (W) = 2.629
    Fitted to optimize data collapse in Eq (19); affects extracted d.
  • Correction exponent delta in ansatz (22) per model and distance = 0.16 to 0.57 (Table 4)
    Phenomenological parameter in the finite-size extrapolation.
  • Constants a and b in ansatz (22) per model and distance = a approximately 0.98 to 1.00, b approximately -0.003 to 0.021 (Table 4)
    Phenomenological parameters in the finite-size extrapolation.
  • DLFPP analysis parameters nu and s = nu in {0, 0.2, ..., 0.8}, s in {0, 2, 4}
    Varied to estimate systematic error in lambda.
assumptions (5)
  • domain assumption Pointwise scaling limit (19) for distance distributions
    Assumed to hold for all four planar map models; known only for uniform quadrangulations; not implied by (18).
  • ad hoc to paper Finite-size correction ansatz (22) with fitted a, b, delta
    Phenomenological form used to extrapolate k_n to large n; no derivation.
  • domain assumption Universality: each planar map model converges to Liouville quantum gravity with the stated gamma
    Based on prior works [25,30,31,32]; the paper relies on this to map c to gamma.
  • domain assumption Discrete LFPP distance approximates continuum LFPP (Eq 36)
    From [29, Theorem 1.4/1.5]; used to interpret lattice simulations as continuum LQG measurements.
  • standard math KPZ relation c = 25 - 6(2/gamma + gamma/2)^2
    Standard result from Liouville CFT, used to translate between c and gamma.

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Cite this review

Pith. "Pith review of Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity." pith.science (2026). https://pith.science/paper/OIUJE4HF

@misc{pith2026190809469,
  author       = {Pith},
  title        = {Pith review of: Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIUJE4HF}},
  note         = {Machine review of arXiv:1908.09469}
}
abstract

Two-dimensional quantum gravity, defined either via scaling limits of random discrete surfaces or via Liouville quantum gravity, is known to possess a geometry that is genuinely fractal with a Hausdorff dimension equal to 4. Coupling gravity to a statistical system at criticality changes the fractal properties of the geometry in a way that depends on the central charge of the critical system. Establishing the dependence of the Hausdorff dimension on this central charge $c$ has been an important open problem in physics and mathematics in the past decades. All simulation data produced thus far has supported a formula put forward by Watabiki in the nineties. However, recent rigorous bounds on the Hausdorff dimension in Liouville quantum gravity show that Watabiki's formula cannot be correct when $c$ approaches $-\infty$. Based on simulations of discrete surfaces encoded by random planar maps and a numerical implementation of Liouville quantum gravity, we obtain new finite-size scaling estimates of the Hausdorff dimension that are in clear contradiction with Watabiki's formula for all simulated values of $c\in (-\infty,0)$. Instead, the most reliable data in the range $c\in [-12.5, 0)$ is in very good agreement with an alternative formula that was recently suggested by Ding and Gwynne. The estimates for $c\in(-\infty,-12.5)$ display a negative deviation from the latter formula, but the scaling is seen to be less accurate in this regime.

Figures

Figures reproduced from arXiv: 1908.09469 by the authors.

Figure 1
Figure 1. Three-dimensional visualizations of random surfaces sampled from three of the models under consideration: (a) spanning-tree-decorated quadrangulations, (b) bipolar-oriented triangulations, (c) Schnyder-wood￾decorated triangulations. Each one consists of 215 faces and is coloured (from red to blue) by increasing graph distance to a randomly sampled vertex. arXiv:1908.09469v1 [gr-qc] 26 Aug 2019 [PITH_FULL_IMAGE:figu… view at source ↗
Figure 2
Figure 2. (a) The known bounds [28, 25, 29] on the Hausdorff dimension dγ together with the predictions by Watabiki and Ding & Gwynne. (b) A zoomed-in version shifted by Watabiki’s prediction together with numerical estimates: [17] in brown, [19] in purple, [21] in orange, [22] in black, [23] in green. (ii) The numerical estimate of the Hausdorff dimension in random triangulations is inaccurate. (iii) The actual Hausdorff dim… view at source ↗
Figure 3
Figure 3. (a) A rooted quadrangulation in Mp4q 10 . (b) A decoration by spanning trees. (c) The corresponding tN, W, S, Eu-walk in the quadrant: N-E-E-S-W-N-E￾N-E-N-W-E-S-S-S-N-W-W-W-S. (d) The schematic operations corresponding to the steps of the walk. (e) The first seven steps in the construction. With the bijection in hand we can generate a random quadrangulation according to the partition function (11) from a uniform ran… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: (a) A triangulation in Mp3q 10 with the root in green. (b) A bipolar orientation (all edges oriented to the north). (c) The corresponding tN, W, SEu￾walk in the quadrant: N-N-SE-SE-N-W-SE-N-W-N-W-SE-SE-W-W. (d) The schematic operations corresponding to the steps of the…
Figure 5
Figure 5. Figure 5: (a) A simple triangulation in Mp3q 12 . (b) The condition on the ordering of edges around inner vertices. (c) A Schnyder wood. (d) The corresponding tE, W, NW, SEu-walk of length 10: E-E-W-NW-E-W-E-SE-W-W. (e) The corresponding double Dyck path. (f) The red tree togeth…
Figure 6
Figure 6. Figure 6: The graph distance (left) and dual graph distance (right) in a quadrangulation [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Finite-size scaling for the dual graph distance in bipolar-oriented triangulations without shift (left) and with shift s “ 2.941 (right). The effect in the case of graph distance is less pronounced. We estimate ρ p˚q n with high accuracy for each model p˚q and sizes n …
Figure 8
Figure 8. Figure 8: Finite-size scaling of the graph distance in the four models with sizes ranging from 210 “ 1024 to 224 “ 16 777 216 and shifts as listed in the table. The collapse of ρ pBq: n in the case of dual graph distance on bipolar-oriented triangulations is shown in the left pl…
Figure 9
Figure 9. Figure 9: Plots of logpn0{nq logpkn{kn0 q with statistical error bars for all four models, showing both graph distance (purple) and dual graph distance (green) measurements. Watabiki’s and Ding-Gwynne’s predictions are indicated by horizontal lines. With s fixed we determine the…
Figure 10
Figure 10. Figure 10: Plots of knpn{n0q 1{dWγ with the scaling parameters kn established via finite-size scaling of the graph distance (purple) and dual graph distance (green). The solid curves correspond to best fits of the ansatz (22) [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: The plot on the left shows the estimates of dγ in comparison to dW γ in the same scale as Figure 2b. The right plot shows the individual estimates of dγ using the graph distance (purple) and dual graph distance (green) in comparison with d DG γ . 4. Hausdorff dimensio…
Figure 12
Figure 12. Figure 12: Top: plot of ξψpxq on Λ2048 together with the shortest DLFPP paths (white) from 152 evenly spaced points to the center of the lattice. Bottom: the DLFPP distance Dξ,wpx, yq from points x P Λ2048 to the center y, colored from blue to red according to the fractional par…
Figure 13
Figure 13. Figure 13: Finite-size scaling of the DLFPP distance for ξ “ 0.1, 0.2, 0.3, 0.4. Shown are plots of x ÞÑ k ´1 w ρξ,wpk ´1 w xq with kw determined by a best fit for w “ 2 7 , . . . , 2 12 [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: The plot on the left shows the estimates of λ with systematic errors (as recorded in [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: Left: The DLFPP estimates of dγ (error bars too small to display). Right: the estimates of dγ normalized by d DG γ with error bars as obtained from DLFPP (black) and random planar maps (green). λ “ ξ ? 6 ` 1 ´ 1 3 p1 ´ ? 6 ξq 3 ˘ . However, we are hesitant to rule out…

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  1. Random surfaces and Liouville quantum gravity

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    An expository overview of the definition of Liouville quantum gravity surfaces, the three senses in which random planar maps converge to them, and the major open problems.

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