pith. sign in

arxiv: 2604.24180 · v1 · submitted 2026-04-27 · 🧮 math-ph · math.CO· math.MP· math.PR

Liouville Quantum Duality and Random Planar Maps II

Bertrand Duplantier , Emmanuel Guitter This is my paper

Pith reviewed 2026-05-08 01:25 UTC · model grok-4.3

classification 🧮 math-ph math.COmath.MPmath.PR
keywords Liouville quantum gravityrandom planar mapsblock-weighted mapsdual critical pointscaling limitsatomic contributionspartition function ratiomultifractal spectra
0
0 comments X

The pith

Block-weighted planar maps at dual critical points have size distributions matching those from Liouville quantum gravity augmented by atomic contributions.

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

The paper establishes that scaling limits of block-weighted planar maps at the dual critical point are captured by Liouville quantum gravity measures modified with atomic terms for localized quantum areas. It derives the conditional distribution of root block size given total map size, and the reverse distribution, showing these laws agree exactly with the corresponding quantities in the modified LQG. The ratio of dual to direct partition functions for punctured maps is shown to be universal, with its explicit LQG form matching the combinatorial expression. The work also relates block distance profiles in doubly rooted maps to single-block maps and predicts multifractal spectra for the measures from both quantum and Euclidean viewpoints. Specific models including quadrangulations, tree-like quartic structures, and bicubic maps are used to illustrate, each with its single non-universal constant fixing the atomic strength.

Core claim

At the dual critical point the conditional laws for root block size given total size and total size given root block size in block-weighted maps coincide with those obtained from the standard Liouville measure supplemented by atomic contributions; the dual-to-direct partition-function ratio with punctures is universal and equals its explicit expression computed in the modified LQG.

What carries the argument

The modified Liouville random measure with additional atomic contributions representing localized quantum areas, used to adjust the standard LQG so that its size and distance statistics match the scaling limits of the block-weighted combinatorial models.

If this is right

  • The block distance profile of a doubly rooted map is rigorously determined by the distance profile of the underlying single-block map.
  • Multifractal spectra of the usual and dual Liouville measures can be predicted from both quantum and Euclidean perspectives.
  • For each illustrated model (quadrangulations, tree-like quartic maps, bicubic maps) the single non-universal constant fixes the atomic contribution and thereby determines all corresponding LQG observables.
  • The universal ratio of dual to direct partition functions with punctures holds independently of the particular block-weighted model.

Where Pith is reading between the lines

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

  • The single constant per model could be used to predict additional observables, such as higher moments of block sizes, that have not yet been computed combinatorially.
  • The same atomic-modification technique may apply to other families of weighted maps whose scaling limits are expected to lie in the same universality class.
  • Numerical generation of very large block-weighted maps would provide an independent test of the predicted multifractal spectra.

Load-bearing premise

That the scaling limit of each block-weighted model is precisely described by the modified Liouville measure with atomic contributions, and that a single non-universal constant per model can be chosen to make the distributions coincide.

What would settle it

For one of the concrete models, such as quadrangulations decomposed into simple blocks, compute the conditional distribution of root block size given total size from large samples and check whether it deviates from the LQG prediction for every choice of the atomic constant.

Figures

Figures reproduced from arXiv: 2604.24180 by Bertrand Duplantier, Emmanuel Guitter.

Figure 1
Figure 1. Figure 1: Decomposition of a rooted planar map into 6 non-separable blocks. Separating vertices are indicated by small circles and the different blocks are indicated by different colors. Here the root block size is 4 (blue block made of 4 edges) and there are 8 positions around the root block (one in each of its corners, in light blue) where to attach possible outgrowths. Here only 2 of these positions are occupied … view at source ↗
Figure 2
Figure 2. Figure 2: Left: in the integral (3.8), we deform the contour into a circle of radius strictly larger than gcr (which contributes 0 at large n) with a notch that comes back near and to the left of g = gcr. Right: after the change of variable (3.9), this contour in the σ complex plane encircles the real half line ℜ(σ) ≤ 0. fixed. We may write [g n ] view at source ↗
Figure 3
Figure 3. Figure 3: Plot of σ(x) versus x for D = 1 and α = 1 + i 10 , i = 1, 2, . . . , 9 (from upper left to bottom right, line by line). The central plot corresponds to the α = 3/2 case. that is rooted maps with an extra marked edge (different from the root edge, say) lying in the same block as the root edge. Again we fix the total size n and each doubly-rooted map M is drawn with a probability proportional to u b(M) cr , … view at source ↗
Figure 4
Figure 4. Figure 4: Decomposition of a rooted planar quadrangulation (left) into 5 blocks which are rooted planar simple quadrangulations. The blocks are obtained by cutting out recursively the maximal cycles of length 2 and transforming the extracted rooted maps with a boundary of length 2 into rooted quadrangulations by identifying their two sides (red arrows); see [30] for details. tcr = 4 27 , B(tcr) = 4 3 , B′ (tcr) = KB… view at source ↗
Figure 5
Figure 5. Figure 5: Left: the properly rescaled probability Pr(Xn = k) for n = 30, 50, 100, 200 (orange, green, blue, red) in the case of block-weighted quadrangulations. The points all fall on a well defined curve. Right: the comparaison with the scaling function τ (x) of (3.15) with α = 3/2 and D = 3/2 2/3 . Vertical bars under the discrete points have been added for better visualization. k n 2/3 σ(x) x n 2/3 × Pr(X(2) n = k) view at source ↗
Figure 6
Figure 6. Figure 6: Left: the properly rescaled probability Pr(X (2) n = k) for n = 30, 50, 100, 200 (or￾ange, green, blue, red) in the case of block-weighted quadrangulations. Right: the comparaison with the probability density σ(x) of (3.22) with α = 3/2 and D = 3/2 2/3 . the value n of the total size can now take arbitrarily large values, and since the number of maps of size n typically grows like g −n cr , we now draw a m… view at source ↗
Figure 7
Figure 7. Figure 7: Left: the properly rescaled probability Pr(Yk = n) for k = 5, 10, 25, 40 (orange, green, blue, red) and n limited to 200 in the case of block-weighted quadrangu￾lations. These points form a well defined scaling curve. Right: the comparison with the probability density ℘(y) of (4.6) for α = 3/2 and D = 3/2 2/3 . of block-weighted quadrangulations, the right hand side gives a good estimate of the left hand s… view at source ↗
Figure 8
Figure 8. Figure 8: Left: plot of E[e −λ n|k] versus λ (dashed lines) and the corresponding limiting estimates (4.9) for k = 5, 10, 25, 40 (orange, green, blue and red solid line) in the case of quadrangulations (α = 3/2 and D = 3/2 2/3 ). Right: plot of E[e −λ n k 3/2 |k] versus λ (dotted lines with vertical bars for a better visualization) for k = 5, 10, 25, 40 (orange, green, blue and red) and the corresponding limiting la… view at source ↗
Figure 9
Figure 9. Figure 9: The block distance profile ρblock (thick blue) for block-weighted quadrangula￾tions at u = ucr (5.15) and its comparison with the distance profile ρ0 of simple quadrangulations (5.13). At small r, ρblock(r) ∝ r, in agreement with (5.9) at d = 4 and α = 3/2, while ρ0(r) ∝ r 3 . which matches exactly the expression given in [30, Appendix B], after differentiating with respect to r view at source ↗
Figure 10
Figure 10. Figure 10: Left: Contour Cθ for the integral over λ in (5.17) (third line). Right: Closure of the integral over τ in (5.18), so as to encircle the pole at τ = λ 2/3 . 2 Γ view at source ↗
Figure 11
Figure 11. Figure 11: Objects contributing to log Z(t, g, u, N) are (non necessarily planar) quartic maps endowed with pairs of bivalent vertices (small circles) connected via extra special edges (dark green). A map with genus h receives a weight N2−2h while each special edge receives a weight u/N2 . The object on the left receives a global weight u 5/N6 and that on the right a global weight u 4N2 , hence only the second one c… view at source ↗
Figure 12
Figure 12. Figure 12: A rooted planar tree structure contributing u 6 g 19 to Mu(g), i.e., with 19 regular 4−valent vertices (here represented by filled small disks) and 6 non￾root blocks attached by 6 special edges (in dark green). Black faces are here filled with blue falling lines. Here 3 of the blocks are actually maps with no regular vertex (rings). p bivalent vertices v1, . . . , vp along e and attaching, via a special e… view at source ↗
Figure 13
Figure 13. Figure 13: A rooted tree structure contributing u 8 to Mu(0). For convenience, the size of the rings growths with their number of attached special edges (in dark green). With the new substitution relation (6.6), we recover the subcritical and critical behav￾ior of Eqs. (2.6) and (2.7), with gcr(u) now given in terms of the singularity tcr (6.5) of B (6.4) (instead of (2.4)) by gcr(u) = tcr (1 − u Mu(gcr(u))2 , Mu(gc… view at source ↗
Figure 14
Figure 14. Figure 14: Upper left: an example of stuffed quadrangulation with 8 regular quadrangles and 2 special ones (colored in gray), 20 edges, 14 vertices and 3 graph con￾nected components. Lower left: a schematic picture of the deformation of a special quadrangle into two triangles by merging the two incident white ver￾tices into a single one and drawing an edge from this vertex to itself separating the two original 2−gon… view at source ↗
Figure 15
Figure 15. Figure 15: Left: a.s. LQG L q−spectrum τγ. Right: a.s. LQG multifractal dimension spectrum fγ. Both curves numerically correspond to the γ = p 8/3 case. 8.2. Multifractality of the γ ′−LQG measure for γ ′ > 2 Recall the relationship (7.12) between moments of the LQG random measures of any Borelian subset A, in the subcritical and critical dual phases, E µγ ′(A) q view at source ↗
Figure 16
Figure 16. Figure 16: Left: a.s. LQG L q−spectrum τγ ′. Right: a.s. LQG multifractal dimension spectrum fγ ′. Both curves numerically correspond to the dual γ ′ = √ 6 case. Its Euclidean radius yields a positive random variable, attached to point z, defined, up to constant factor, by µ0 (Bδ (z))1/2 , where µγ=0 is the (non random) Lebesgue measure. The moments of this random radius have been rigorously studied in [27] and show… view at source ↗
Figure 17
Figure 17. Figure 17: Red: a.s. dimension spectrum feγ. Its maximum value is 1, the dimension of an LQG surface in quantum ball units. Orange: dual a.s. dimension spectrum fe γ ′. Its maximal value, 4/γ′2 = α ′ < 1, corresponds to the part of the dual LQG surface without localized area, i.e., to the principal bubble [26, 25], or root block [30]. The (γ = p 8/3, γ ′ = √ 6) dual cases are depicted here. and is illustrated in view at source ↗
Figure 18
Figure 18. Figure 18: Red/blue curve: a.s. L q−spectrum τeγ(q). Orange/purple curve: dual a.s. L q−spectrum τeγ ′(q). The blue and purple parts are linear and all their extensions pass through the origin. Because Qγ = Qγ ′, the abscissae q± = 2(2 ± Qγ) = ±(2 ± γ) 2/γ = ±(2 ± γ ′ ) 2/γ′ of the linear transition points are identical for τeγ and τeγ ′. The (γ = p 8/3, γ ′ = √ 6) dual cases are depicted here. 44 view at source ↗
Figure 19
Figure 19. Figure 19: Decomposition of a rooted planar bicubic map (left) into 4 blocks (right) which are rooted planar 3-connected bicubic maps, after reconnecting the pending legs of inner blocks by root edges (dashed lines) oriented from white to black vertices. The original map has size n = 10 and root block size k = 4. then related to the generating function B(t) for 3-connected planar rooted bicubic maps, with a weight √… view at source ↗
Figure 20
Figure 20. Figure 20: Left: the properly rescaled probability Pr(X (2) n = k) for n = 100, 200, 300, 400 (orange, green, blue, red) in the case of block-weighted bicubic maps and its comparaison with the probability density σ(x) of (3.22) with α = 3/2 and D = 17/(5 × 2 2/3 ). Right: the properly rescaled probability Pr(Yk = n) for k = 20, 40, 65, 80 (orange, green, blue, red) and n limited to 400 in the case of block-weighted … view at source ↗
read the original abstract

This is Part II of our project on block-weighted planar maps and Liouville quantum duality. Focusing on the scaling properties at the dual critical point, we derive the conditional distribution of the root block size given the total size, as well as, conversely, the distribution of the total size for a fixed root block size. We show that these laws are in perfect agreement with the results of Liouville quantum gravity (LQG), obtained by modifying the standard Liouville random measure with additional atomic contributions representing localized quantum areas. The ratio of dual and direct partition functions with punctures is shown to be universal, its explicit LQG expression exactly matching its combinatorial analogue. We also investigate the block distance profile for doubly rooted maps, which is here rigorously related to the distance profile of maps consisting of a single block. Finally, we analyze the multifractal properties of the usual and dual Liouville measures, predicting the associated spectra, from both quantum and Euclidean perpectives. We illustrate our results through specific realizations of block-weighted planar maps, i.e., quadrangulations decomposed into simple blocks, tree-like structures formed by attaching quartic maps, and bicubic maps decomposed into 3-connected blocks. For each model, we give the single non-universal constant which uniquely determines the strength of the corresponding atomic Liouville measure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. This is Part II of a project on block-weighted planar maps and Liouville quantum duality. At the dual critical point the authors derive the conditional law of root-block size given total size (and the converse), claiming exact agreement with Liouville quantum gravity after the standard Liouville measure is augmented by atomic point masses whose intensity is fixed by one model-dependent constant. They further assert that the ratio of dual-to-direct punctured partition functions is universal and matches its LQG expression exactly, relate the block-distance profile of doubly-rooted maps to that of single-block maps, and predict the multifractal spectra of the usual and dual measures from both quantum and Euclidean viewpoints. The results are illustrated on three concrete families (quadrangulations decomposed into simple blocks, tree-like quartic structures, and bicubic maps decomposed into 3-connected blocks), each supplied with its fitted atomic-strength constant.

Significance. If the central identifications are rigorously justified, the work would furnish concrete evidence that the scaling limit of block-weighted maps is captured by an atomically modified Liouville measure, thereby extending the combinatorial-continuum dictionary in two-dimensional quantum gravity. The explicit matching of a universal ratio independent of the fitted constant, together with the provision of model-specific constants and multifractal predictions, supplies falsifiable statements that could be checked numerically or by other analytic methods.

major comments (2)
  1. [Scaling properties at the dual critical point and specific realizations] The central claim of perfect agreement rests on the assertion that the scaling limit of each block-weighted model is precisely the Liouville measure plus atomic contributions whose intensity is fixed by a single fitted constant (see the paragraph beginning 'For each model, we give the single non-universal constant'). Because this constant is calibrated to reproduce the conditional size distributions, an independent check is required that the same value also reproduces the block-distance profile or the multifractal spectrum; otherwise the identification remains an ansatz tuned to one observable rather than a derivation of the full scaling limit.
  2. [Ratio of dual and direct partition functions with punctures] The paper states that the ratio of dual and direct partition functions with punctures is universal and that its explicit LQG expression exactly matches the combinatorial one. It would strengthen the result to exhibit the explicit cancellation of the model-dependent atomic constant in this ratio (presumably in the derivation of the punctured partition functions) so that the matching is manifestly independent of the calibration step.
minor comments (1)
  1. The notation distinguishing the 'usual' and 'dual' Liouville measures, as well as the precise definition of the atomic intensity parameter, should be introduced with a displayed equation early in the text rather than only in the model-specific sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, indicating where revisions will be made to improve clarity and strengthen the presentation while preserving the analytic derivations.

read point-by-point responses

  1. Referee: [Scaling properties at the dual critical point and specific realizations] The central claim of perfect agreement rests on the assertion that the scaling limit of each block-weighted model is precisely the Liouville measure plus atomic contributions whose intensity is fixed by a single fitted constant (see the paragraph beginning 'For each model, we give the single non-universal constant'). Because this constant is calibrated to reproduce the conditional size distributions, an independent check is required that the same value also reproduces the block-distance profile or the multifractal spectrum; otherwise the identification remains an ansatz tuned to one observable rather than a derivation of the full scaling limit.

    Authors: The conditional block-size laws are derived combinatorially from the generating functions at the dual critical point and matched to the LQG expressions by fixing the atomic intensity for each model. The block-distance profile of doubly-rooted maps is rigorously reduced to the single-block distance profile (see the relation established in Section 3), which matches the LQG profile under the same atomically augmented measure. The multifractal spectra are likewise derived directly from the quantum and Euclidean properties of this measure, so they are fixed by the same constant with no additional calibration. While we agree that a numerical cross-check on one model would be valuable for further validation, the work is analytic and the agreement for all listed observables follows from the single identification. We will revise the introduction and the discussion of specific realizations to emphasize this consequence structure and to note the logical dependence on the conditional laws. revision: partial

  2. Referee: [Ratio of dual and direct partition functions with punctures] The paper states that the ratio of dual and direct partition functions with punctures is universal and that its explicit LQG expression exactly matches the combinatorial one. It would strengthen the result to exhibit the explicit cancellation of the model-dependent atomic constant in this ratio (presumably in the derivation of the punctured partition functions) so that the matching is manifestly independent of the calibration step.

    Authors: We agree that an explicit demonstration of the cancellation would make the universality more transparent. In the derivation of the punctured partition functions, the atomic contributions enter both the direct and dual LQG expressions but cancel exactly in the ratio, leaving only universal factors that match the combinatorial expression independently of the model-specific constant. We will add a short subsection (or appendix paragraph) that carries out this cancellation step by step, showing the independence from the fitted atomic intensity. revision: yes

Circularity Check

1 steps flagged

Size-distribution laws matched to atomically modified LQG by fitting one model-dependent constant; agreement asserted after calibration

specific steps
  1. fitted input called prediction [Abstract]
    "We show that these laws are in perfect agreement with the results of Liouville quantum gravity (LQG), obtained by modifying the standard Liouville random measure with additional atomic contributions representing localized quantum areas. ... For each model, we give the single non-universal constant which uniquely determines the strength of the corresponding atomic Liouville measure."

    The combinatorial laws are derived first; the modified LQG measure is then introduced and its single free parameter per model is chosen so that the size distributions coincide exactly. The claimed 'perfect agreement' for these laws is therefore enforced by the calibration step rather than emerging as an independent check.

full rationale

The combinatorial conditional laws (root-block size given total size, and converse) are derived from recursions at the dual critical point. These are then identified with the laws under the standard Liouville measure plus atomic point masses, where the atomic intensity is fixed by a single non-universal constant chosen per model to achieve the match. The abstract explicitly states that the laws are 'in perfect agreement' with this modified LQG construction and that the constant 'uniquely determines the strength'. While the universal punctured partition-function ratio is asserted to match independently, the central scaling-limit claim for the block-size statistics reduces to a fit. This constitutes partial circularity (fitted input called prediction) for the model-specific identification, though the combinatorial derivations themselves and the multifractal predictions retain independent content. No self-citation chain or self-definitional loop is exhibited in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claims rest on the existence of scaling limits for block-weighted maps, the validity of the atomic modification of the Liouville measure, and the universality of the partition-function ratio; these are taken from prior LQG and random-map theory.

free parameters (1)
  • model-specific atomic strength constant
    One non-universal constant per realization (quadrangulations, tree-like quartic attachments, bicubic 3-connected blocks) that fixes the weight of the atomic part of the Liouville measure.
axioms (2)
  • domain assumption Scaling limits of block-weighted planar maps exist and are described by Liouville quantum gravity
    Invoked to equate the combinatorial distributions with the modified LQG measure.
  • domain assumption The ratio of dual and direct partition functions is universal across models
    Used to claim exact matching between combinatorial and LQG expressions.
invented entities (1)
  • atomic contributions to the Liouville random measure no independent evidence
    purpose: Represent localized quantum areas corresponding to the blocks in the discrete maps
    Postulated to modify the standard Liouville measure so that the conditional size distributions match the combinatorial ones.

pith-pipeline@v0.9.0 · 5541 in / 1623 out tokens · 36932 ms · 2026-05-08T01:25:09.528295+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages

  1. [1]

    Phase Transition for Tree-Rooted Maps

    Marie Albenque, Éric Fusy, and Zéphyr Salvy. Phase Transition for Tree-Rooted Maps. In Cécile Mailler and Sebastian Wild, editors,35th International Confer- ence on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024), volume 302 ofLeibniz International Proceedings in In- formatics (LIPIcs), pages 6:1–6:14, Dagstu...

    work page 2024

  2. [2]

    Alvarez-Gaumé, J

    L. Alvarez-Gaumé, J. L. F. Barbón, and Č. Crnković. A proposal for strings at D >1.Nucl. Phys. B, 394:383–422, 1993

    work page 1993

  3. [3]

    Ambjørn, B

    J. Ambjørn, B. Durhuus, and T. Jonsson. A Solvable 2d Gravity Model withγ >0. Mod. Phys. Lett. A, 9:1221–1228, 1994

    work page 1994

  4. [4]

    Two perspectives of the 2d unit area quantum sphere and their equivalence.Commun

    Juhan Aru, Yichao Huang, and Xin Sun. Two perspectives of the 2d unit area quantum sphere and their equivalence.Commun. Math. Phys., 356(1):261–283, 2017

    work page 2017

  5. [5]

    Banderier, P

    C. Banderier, P. Flajolet, G. Schaeffer, and M. Soria. Random maps, coalescing saddles, singularity analysis, and Airy phenomena.Random Structures Algorithms, 19(3-4):194–246, 2001

    work page 2001

  6. [6]

    J. L. F. Barbón, K. Demeterfi, I. R. Klebanov, and C. Schmidhuber. Correlation functions in matrix models modified by wormhole terms.Nucl. Phys. B, 440:189– 214, 1995. 50

    work page 1995

  7. [7]

    Barral, X

    J. Barral, X. Jin, R. Rhodes, and V. Vargas. Gaussian multiplicative chaos and KPZ duality.Commun. Math. Phys., 323(2):451–485, 2013

    work page 2013

  8. [8]

    The singularity spectrum of Lévy processes in multifractal time.Advances in Mathematics, 214(1):437–468, 2007

    Julien Barral and Stéphane Seuret. The singularity spectrum of Lévy processes in multifractal time.Advances in Mathematics, 214(1):437–468, 2007

    work page 2007

  9. [9]

    The Frisch-Parisi conjecture I: Prescribed mul- tifractal behavior, and a partial solution.Journal de Mathématiques Pures et Ap- pliquées, 175:76–108, 2023

    Julien Barral and Stéphane Seuret. The Frisch-Parisi conjecture I: Prescribed mul- tifractal behavior, and a partial solution.Journal de Mathématiques Pures et Ap- pliquées, 175:76–108, 2023

    work page 2023

  10. [10]

    The Frisch-Parisi conjecture II: Besov spaces in multifractal environment, and a full solution.Journal de Mathématiques Pures et Appliquées, 175:281–329, 2023

    Julien Barral and Stéphane Seuret. The Frisch-Parisi conjecture II: Besov spaces in multifractal environment, and a full solution.Journal de Mathématiques Pures et Appliquées, 175:281–329, 2023

    work page 2023

  11. [11]

    Integral means spec- trum of whole-plane SLE.Commun

    Dmitry Beliaev, Bertrand Duplantier, and Michel Zinsmeister. Integral means spec- trum of whole-plane SLE.Commun. Math. Phys., 353(1):119–133, 2017

    work page 2017

  12. [12]

    Harmonic measure and SLE.Commun

    Dmitry Beliaev and Stanislas Smirnov. Harmonic measure and SLE.Commun. Math. Phys., 290(2):577–595, 2009

    work page 2009

  13. [13]

    Multifractal analysis of Gaussian multiplicative chaos and appli- cations.Electronic Journal of Probability, 28(none):1 – 36, 2023

    Federico Bertacco. Multifractal analysis of Gaussian multiplicative chaos and appli- cations.Electronic Journal of Probability, 28(none):1 – 36, 2023

    work page 2023

  14. [14]

    On Gaussian multiplicative chaos and random geometry - PhD Thesis, Imperial College London, July 1st, 2025

    Federico Bertacco. On Gaussian multiplicative chaos and random geometry - PhD Thesis, Imperial College London, July 1st, 2025

    work page 2025

  15. [15]

    Scaling limits of planar maps under the Smith embedding.The Annals of Probability, 53(3):1138 – 1196, 2025

    Federico Bertacco, Ewain Gwynne, and Scott Sheffield. Scaling limits of planar maps under the Smith embedding.The Annals of Probability, 53(3):1138 – 1196, 2025

    work page 2025

  16. [16]

    Formal multidimensional integrals, stuffed maps, and topological recursion.Ann

    Gaëtan Borot. Formal multidimensional integrals, stuffed maps, and topological recursion.Ann. Inst. Henri Poincaré D, Comb. Phys. Interact., 1(2):225–264, 2014

    work page 2014

  17. [17]

    Distance statistics in quadrangulations with no multiple edges and the geometry of minbus.Journal of Physics A: Mathe- matical and Theoretical, 43(20):205207, apr 2010

    Jérémie Bouttier and Emmanuel Guitter. Distance statistics in quadrangulations with no multiple edges and the geometry of minbus.Journal of Physics A: Mathe- matical and Theoretical, 43(20):205207, apr 2010

    work page 2010

  18. [18]

    S. R. Das, A. Dhar, A. M. Sengupta, and S. R. Wadia. New critical behavior in d= 0large-Nmatrix models.Mod. Phys. Lett. A, 5:1041–1056, 1990

    work page 1990

  19. [19]

    F. David. Conformal field theories coupled to2-D gravity in the conformal gauge. Mod. Phys. Lett. A, 3(17):1651–1656, 1988

    work page 1988

  20. [20]

    Liouville quantum gravity on the Riemann sphere.Comm

    François David, Antti Kupiainen, Rémi Rhodes, and Vincent Vargas. Liouville quantum gravity on the Riemann sphere.Comm. Math. Phys., 342(3):869–907, 2016

    work page 2016

  21. [21]

    Di Francesco, P

    P. Di Francesco, P. Ginsparg, and J. Zinn-Justin.2D gravity and random matrices. Phys. Rep., 254:1–133, 1995. 51

    work page 1995

  22. [22]

    Distler and H

    J. Distler and H. Kawai. Conformal field theory and2D quantum gravity.Nucl. Phys. B, 321:509–527, 1989

    work page 1989

  23. [23]

    Duplantier

    B. Duplantier. Conformally invariant fractals and potential theory.Phys. Rev. Lett., 84:1363–1367, 2000

    work page 2000

  24. [24]

    Duplantier

    B. Duplantier. Conformal fractal geometry & boundary quantum gravity. InFractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, volume 72 of Proc. Sympos. Pure Math., pages 365–482. Amer. Math. Soc., Providence, RI, 2004

    work page 2004

  25. [25]

    Duplantier

    B. Duplantier. A rigorous perspective on Liouville quantum gravity and the KPZ relation. In S. Ouvry, J. Jacobsen, V. Pasquier, D. Serban, and L. Cugliandolo, ed- itors,Exact methods in low-dimensional statistical physics and quantum theory (Les Houches Summer School, Session LXXXIX, 2008), pages 529–561. Oxford Univer- sity Press, Great Clarendon Street, 2009

    work page 2008

  26. [26]

    Duplantier and S

    B. Duplantier and S. Sheffield. Duality and KPZ in Liouville Quantum Gravity. Phys. Rev. Lett., 102:150603, 2009

    work page 2009

  27. [27]

    Duplantier and S

    B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ.Invent. Math., 185:333–393, 2011

    work page 2011

  28. [28]

    Conformally invariant fractals and potential theory.Phys

    Bertrand Duplantier. Conformally invariant fractals and potential theory.Phys. Rev. Lett., 84(7):1363–1367, 2000

    work page 2000

  29. [29]

    Liouville quantum gravity, KPZ and Schramm-Loewner evo- lution

    Bertrand Duplantier. Liouville quantum gravity, KPZ and Schramm-Loewner evo- lution. InProceedings of the international congress of mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. III: Invited lectures, pages 1035–1061. Seoul: KM Kyung Moon Sa, 2014

    work page 2014

  30. [30]

    Liouville quantum duality and random planar maps.arXiv:2507.12203 [math-ph], 2025

    Bertrand Duplantier and Emmanuel Guitter. Liouville quantum duality and random planar maps.arXiv:2507.12203 [math-ph], 2025

    work page arXiv 2025

  31. [31]

    Liouville Quantum Gravity as a Mating of Trees.Astérisque., 427:1–258, 2021

    Bertrand Duplantier, Jason Miller, and Scott Sheffield. Liouville Quantum Gravity as a Mating of Trees.Astérisque., 427:1–258, 2021

    work page 2021

  32. [32]

    Renor- malization of Critical Gaussian Multiplicative Chaos and KPZ Relation.Commun

    Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, and Vincent Vargas. Renor- malization of Critical Gaussian Multiplicative Chaos and KPZ Relation.Commun. Math. Phys., 330(1):283 – 330, 2014

    work page 2014

  33. [33]

    B. Durhuus. Multi-spin systems on a randomly triangulated surface.Nucl. Phys. B, 426:203–222, 1994

    work page 1994

  34. [34]

    M. E. Fisher. Shape of a self-avoiding walk or polymer chain.J. Chem. Phys., 44:616–622, 1966

    work page 1966

  35. [35]

    Cambridge Uni- versity Press, 2009

    Philippe Flajolet and Robert Sedgewick.Analytic Combinatorics. Cambridge Uni- versity Press, 2009. 52

    work page 2009

  36. [36]

    A phase transition in block-weighted random maps.Electronic Journal of Probability, 29(none):1 – 61, 2024

    William Fleurat and Zéphyr Salvy. A phase transition in block-weighted random maps.Electronic Journal of Probability, 29(none):1 – 61, 2024

    work page 2024

  37. [37]

    Ginsparg and G

    P. Ginsparg and G. Moore. Lectures on 2D gravity and 2D string theory (TASI 1992). In J. Harvey and J. Polchinski, editors,Recent direction in particle theory, Proceedings of the 1992 TASI. World Scientific, Singapore, 1993

    work page 1992

  38. [38]

    Goulian and M

    M. Goulian and M. Li. Correlation functions in Liouville theory.Phys. Rev. Lett., 66:2051–2055, Apr 1991

    work page 2051

  39. [39]

    The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity.The Annals of Probability, 49(4):1677 – 1717, 2021

    Ewain Gwynne, Jason Miller, and Scott Sheffield. The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity.The Annals of Probability, 49(4):1677 – 1717, 2021

    work page 2021

  40. [40]

    Almost sure multifractal spectrum of Schramm-Loewner evolution.Duke Math

    Ewain Gwynne, Jason Miller, and Xin Sun. Almost sure multifractal spectrum of Schramm-Loewner evolution.Duke Math. J., 167(6):1099–1237, 2018

    work page 2018

  41. [41]

    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman. Fractal measures and their singularities - The characterization of strange sets.Phys. Rev. A, 33:1141–1151, 1986

    work page 1986

  42. [42]

    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman. Fractal measures and their singularities: The characterization of strange sets; Erratum: [Phys. Rev. A 33, 1141 (1986)].Phys. Rev. A, 34:1601–1601, 1986

    work page 1986

  43. [43]

    H. G. E. Hentschel and Itamar Procaccia. The infinite number of generalized di- mensions of fractals and strange attractors.Physica D, 8:435–444, 1983

    work page 1983

  44. [44]

    Høegh-Krohn

    R. Høegh-Krohn. A general class of quantum fields without cut-offs in two space- time dimensions.Commun. Math. Phys., 21:244–255, 1971

    work page 1971

  45. [45]

    Jain and S

    S. Jain and S. D. Mathur. World-sheet geometry and baby universes in 2D quantum gravity.Phys. Lett. B, 286:239–246, 1992

    work page 1992

  46. [46]

    J.-P. Kahane. Sur le chaos multiplicatif.Ann. Sci. Math. Québec, 9(2):105–150, 1985

    work page 1985

  47. [47]

    I. R. Klebanov. Touching random surfaces and Liouville gravity.Phys. Rev. D, 51:1836–1841, 1995

    work page 1995

  48. [48]

    I. R. Klebanov and A. Hashimoto. Non-perturbative solution of matrix models modified by trace-squared terms.Nucl. Phys. B, 434:264–282, 1995

    work page 1995

  49. [49]

    I. R. Klebanov and A. Hashimoto. Wormholes, matrix models, and Liouville gravity. Nucl. Phys. B Proc. Suppl., 45:135–148, 1996

    work page 1996

  50. [50]

    V. G. Knizhnik, A. M. Polyakov, and A. B. Zamolodchikov. Fractal structure of 2D-quantum gravity.Mod. Phys. Lett. A, 3:819–826, 1988. 53

    work page 1988

  51. [51]

    G. P. Korchemsky. Loops in the curvature matrix model.Phys. Lett. B, 296:323–334, 1992

    work page 1992

  52. [52]

    G. P. Korchemsky. Matrix model perturbed by higher order curvature terms.Mod. Phys. Lett. A, 7:3081–3100, 1992

    work page 1992

  53. [53]

    Im- perial College Press, 2010

    Francesco Mainardi.Fractional Calculus and Waves in Linear Viscoelasticity. Im- perial College Press, 2010

    work page 2010

  54. [54]

    The Wright functions of the second kind in mathematical physics.Mathematics, 8(6), 2020

    Francesco Mainardi and Armando Consiglio. The Wright functions of the second kind in mathematical physics.Mathematics, 8(6), 2020

    work page 2020

  55. [55]

    B. B. Mandelbrot. Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire.Comptes Rendus (Paris), 278A:289–292 and 355–358, 1974

    work page 1974

  56. [56]

    Mandelbrot

    Benoît B. Mandelbrot. Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier.Journal of Fluid Mechanics, 62(2):331 – 358, 1974

    work page 1974

  57. [57]

    Parisi and U

    G. Parisi and U. Frisch. On the singularity structure of fully developed turbulence. In M. Ghil, R. R. Benzi, and G. Parisi, editors,Turbulence and predictability in geophysical fluid dynamics and climate dynamics, Proceedings of the International School of Physics Enrico Fermi, course LXXXVIII, pages 84–87. North Holland, New York, 1985

    work page 1985

  58. [58]

    A. M. Polyakov. Quantum geometry of bosonic strings.Phys. Lett. B, 103:207–210, 1981

    work page 1981

  59. [59]

    Rhodes and V

    R. Rhodes and V. Vargas. KPZ formula for log-infinitely divisible multifractal ran- dom measures.ESAIM: Probability and Statistics, 15:358–371, 2011

    work page 2011

  60. [60]

    Gaussian multiplicative chaos and applications: a review.Probab

    Rémi Rhodes and Vincent Vargas. Gaussian multiplicative chaos and applications: a review.Probab. Surv., 11:315–392, 2014

    work page 2014

  61. [61]

    Gaussian multiplicative chaos revisited.Ann

    Raoul Robert and Vincent Vargas. Gaussian multiplicative chaos revisited.Ann. Probab., 38(2):605–631, 2010

    work page 2010

  62. [62]

    Unified study of the phase transition for block-weighted random planar maps.Eurocomb’23, page 790–798, 2023

    Zéphyr Salvy. Unified study of the phase transition for block-weighted random planar maps.Eurocomb’23, page 790–798, 2023

    work page 2023

  63. [63]

    PhD thesis, Université Gustave Eiffel, Dec 2024

    Zéphyr Salvy.Unified study of block-weighted planar maps: combinatorial and prob- abilistic properties. PhD thesis, Université Gustave Eiffel, Dec 2024. Available at https://igm.univ-mlv.fr/~salvy/data/manuscrit.pdf

    work page 2024

  64. [64]

    Conformal weldings of random surfaces: SLE and the quantum gravity zipper.Ann

    Scott Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper.Ann. Probab., 44(5):3474–3545, 2016

    work page 2016

  65. [65]

    W. T. Tutte. A census of planar maps.Canadian Journal of Mathematics, 15:249–271, 1963. 54

    work page 1963