pith. sign in

arxiv: 2507.13269 · v2 · submitted 2025-07-17 · 🧮 math.PR · math-ph· math.CV· math.MP

Two-sided heat kernel bounds for sqrt{8/3}-Liouville Brownian motion

Sebastian Andres , Naotaka Kajino , Konstantinos Kavvadias , Jason Miller This is my paper

Pith reviewed 2026-05-19 04:27 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.CVmath.MP
keywords Liouville Brownian motionLiouville quantum gravityheat kernel boundsrandom metricsdiffusion processesmetric measure spacestime change
0
0 comments X

The pith

The heat kernel of Liouville Brownian motion at γ=√(8/3) obeys two-sided bounds given by the √(8/3)-LQG metric up to a polylog factor in the exponential.

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

The paper establishes matching upper and lower bounds on the transition densities of Liouville Brownian motion precisely when the parameter γ equals the square root of eight over three. These bounds are expressed directly in terms of the distance induced by the associated Liouville quantum gravity metric. The estimates are optimal apart from a slowly growing polylogarithmic correction inside the exponential. Such quantitative control determines how quickly the diffusion spreads across the random fractal surface and therefore governs many of its geometric and dynamical features.

Core claim

We establish upper and lower bounds for the heat kernel for LBM when γ=√(8/3) in terms of the √(8/3)-LQG metric which are sharp up to a polylogarithmic factor in the exponential.

What carries the argument

The √(8/3)-LQG metric, which supplies the distance function against which the heat kernel of the time-changed Brownian motion is compared after verifying regularity and doubling properties.

If this is right

  • The diffusion spreads at a rate governed by the random LQG distance rather than Euclidean distance.
  • Standard heat-kernel estimates for metric measure spaces apply once the time change and doubling properties are accounted for.
  • Analytic consequences such as Hölder continuity of sample paths and bounds on hitting probabilities follow directly from the kernel estimates.

Where Pith is reading between the lines

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

  • The same comparison strategy may produce bounds for other admissible values of γ once the corresponding metric regularity is verified.
  • The estimates supply a route to studying path intersections and the Hausdorff dimension of the range of LBM.
  • The results link the probabilistic construction of the process to the metric geometry of the underlying quantum surface.

Load-bearing premise

Liouville Brownian motion can be realized as a time change of ordinary Brownian motion and the LQG metric satisfies the regularity and volume-doubling conditions needed for standard heat-kernel comparison arguments.

What would settle it

A numerical computation on a fine discretization of the √(8/3)-LQG surface that shows the ratio of the true heat kernel to the metric-based prediction exceeds every polylog factor for large enough time and distance.

read the original abstract

Liouville Brownian motion (LBM) is the canonical diffusion process on a Liouville quantum gravity (LQG) surface. In this work, we establish upper and lower bounds for the heat kernel for LBM when $\gamma=\sqrt{8/3}$ in terms of the $\sqrt{8/3}$-LQG metric which are sharp up to a polylogarithmic factor in the exponential.

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

1 major / 2 minor

Summary. The paper establishes two-sided upper and lower bounds on the heat kernel of Liouville Brownian motion (LBM) for the parameter γ = √(8/3), expressed in terms of the corresponding √(8/3)-Liouville quantum gravity (LQG) metric. The bounds are claimed to be sharp up to a polylogarithmic factor inside the exponential. The construction proceeds by viewing LBM as a time-changed Euclidean Brownian motion with respect to the LQG measure and then invoking heat-kernel comparison theorems on the metric measure space (d_γ, μ_γ).

Significance. If the stated bounds hold, the result supplies nearly sharp control on the transition densities of the canonical diffusion on a random surface whose geometry is governed by the Brownian map. This is a concrete advance in the analytic study of LQG, as heat-kernel estimates are a basic tool for understanding recurrence, mixing, and spectral properties on these spaces. The work correctly exploits the known doubling and volume-growth properties of the √(8/3)-Brownian map rather than deriving them anew.

major comments (1)
  1. The manuscript invokes the doubling property and Poincaré inequality for the LQG metric measure space without an explicit reference to the precise theorem (or section) in the Brownian-map literature that supplies the constants needed for the heat-kernel comparison. A short paragraph clarifying which result from the existing theory is applied would strengthen the argument.
minor comments (2)
  1. The abstract states that the bounds are 'sharp up to a polylogarithmic factor in the exponential,' but the precise form of the polylog (e.g., whether it is (log(1/t))^C or (log log(1/t))^C) is not displayed in the abstract; adding the explicit expression would improve readability.
  2. Notation for the LQG metric d_γ and the measure μ_γ is introduced without a forward reference to the section where the construction is recalled; a single sentence directing the reader to the relevant background paragraph would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and constructive suggestion. The comment identifies a straightforward way to improve clarity by adding an explicit reference, and we will incorporate this change in the revised version.

read point-by-point responses

  1. Referee: The manuscript invokes the doubling property and Poincaré inequality for the LQG metric measure space without an explicit reference to the precise theorem (or section) in the Brownian-map literature that supplies the constants needed for the heat-kernel comparison. A short paragraph clarifying which result from the existing theory is applied would strengthen the argument.

    Authors: We agree that an explicit citation would strengthen the exposition. In the revised manuscript we will add a short clarifying paragraph (most naturally in the introduction or in the section applying the heat-kernel comparison) that identifies the precise results from the Brownian-map literature supplying the doubling property, volume-growth estimates, and Poincaré inequality for the √(8/3)-LQG metric measure space, together with the constants used in the comparison theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation constructs LBM as a time-changed Euclidean Brownian motion with respect to the LQG measure and applies standard heat-kernel comparison theorems to the metric measure space (d_γ, μ_γ) for γ=√(8/3). The required doubling, Poincaré, and volume-growth properties are invoked from the established theory of the Brownian map, which is external to the present paper. No load-bearing step reduces by the paper's own equations or self-citation chain to a fitted parameter or definition introduced in the same work; the two-sided bounds with polylog factor follow from the comparison arguments without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the prior construction of LBM as a time change of Brownian motion on the LQG surface and on standard analytic properties of heat kernels on metric measure spaces.

axioms (2)
  • domain assumption Liouville Brownian motion exists and is a well-defined diffusion on the LQG surface for γ=√(8/3)
    Invoked to define the process whose heat kernel is being bounded.
  • domain assumption The √(8/3)-LQG metric satisfies volume doubling and other regularity conditions needed for heat kernel estimates
    Required for the comparison arguments that produce the upper and lower bounds.

pith-pipeline@v0.9.0 · 5606 in / 1323 out tokens · 25271 ms · 2026-05-19T04:27:54.236827+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

91 extracted references · 91 canonical work pages

  1. [1]

    D. Aldous. The continuum random tree. I.Ann. Probab., 19(1):1–28, 1991

    work page 1991

  2. [2]

    Andres, D

    S. Andres, D. Croydon, and T. Kumagai. Heat kernel fluctuations for stochastic processes on fractals and random media. InFrom classical analysis to analysis on fractals. Vol. 1. A tribute to Robert Strichartz, Appl. Numer. Harmon. Anal., pages 265–281. Birkh¨auser/Springer, Cham, [2023] ©2023

    work page 2023

  3. [3]

    Andres and N

    S. Andres and N. Kajino. Continuity and estimates of the Liouville heat kernel with applications to spectral dimen- sions.Probab. Theory Related Fields, 166(3-4):713–752, 2016. 130 SEBASTIAN ANDRES, NAOTAKA KAJINO, KONSTANTINOS KAVVADIAS, AND JASON MILLER

    work page 2016

  4. [4]

    M. Ang, N. Holden, and X. Sun. Conformal welding of quantum disks.Electronic Journal of Probability, 28:1–50, 2023

    work page 2023

  5. [5]

    M. Ang, N. Holden, and X. Sun. Integrability of SLE via conformal welding of random surfaces.Comm. Pure Appl. Math., 77(5):2651–2707, 2024

    work page 2024

  6. [6]

    Angel, D

    O. Angel, D. A. Croydon, S. Hernandez-Torres, and D. Shiraishi. Scaling limits of the three-dimensional uniform spanning tree and associated random walk.Ann. Probab., 49(6):3032–3105, 2021

    work page 2021

  7. [7]

    E. Archer. Brownian motion on stable looptrees.Ann. Inst. Henri Poincar ´e Probab. Stat., 57(2):940–979, 2021

    work page 2021

  8. [8]

    J. Aru, Y. Huang, and X. Sun. Two perspectives of the 2D unit area quantum sphere and their equivalence.Comm. Math. Phys., 356(1):261–283, 2017

    work page 2017

  9. [9]

    J. Aru, T. Lupu, and A. Sep ´ulveda. The first passage sets of the 2d gaussian free field: convergence and isomor- phisms.Communications in Mathematical Physics, 375(3):1885–1929, 2020

    work page 1929

  10. [10]

    M. T. Barlow. Random walks on supercritical percolation clusters.Ann. Probab., 32(4):3024–3084, 2004

    work page 2004

  11. [11]

    M. T. Barlow. Analysis on the Sierpinski carpet. InAnalysis and geometry of metric measure spaces, volume 56 of CRM Proc. Lecture Notes, pages 27–53. Amer. Math. Soc., Providence, RI, 2013

    work page 2013

  12. [12]

    M. T. Barlow, D. A. Croydon, and T. Kumagai. Subsequential scaling limits of simple random walk on the two- dimensional uniform spanning tree.Ann. Probab., 45(1):4–55, 2017

    work page 2017

  13. [13]

    Berestycki

    N. Berestycki. Diffusion in planar Liouville quantum gravity.Ann. Inst. Henri Poincar´e Probab. Stat., 51(3):947–964, 2015

    work page 2015

  14. [14]

    Berestycki

    N. Berestycki. An elementary approach to Gaussian multiplicative chaos.Electron. Commun. Probab., 22:Paper No. 27, 12, 2017

    work page 2017

  15. [15]

    Berestycki and E

    N. Berestycki and E. Gwynne. Random walks on mated-CRT planar maps and Liouville Brownian motion.Comm. Math. Phys., 395(2):773–857, 2022

    work page 2022

  16. [16]

    Berestycki and J

    N. Berestycki and J. R. Norris. Lectures on Schramm–Loewner Evolution.Unpublished lecture notes, available at http: // www. statslab. cam. ac. uk/~ james/ Lectures/ sle. pdf, 2016

    work page 2016

  17. [17]

    Berestycki and E

    N. Berestycki and E. Powell. Gaussian free field and Liouville quantum gravity .Preprint, availailable at arXiv:2404.16642, to be published by Cambridge University Press, 2025

    work page arXiv 2025

  18. [18]

    Berestycki, S

    N. Berestycki, S. Sheffield, and X. Sun. Equivalence of Liouville measure and Gaussian free field.Ann. Inst. Henri Poincar´e Probab. Stat., 59(2):795–816, 2023

    work page 2023

  19. [19]

    Bertacco and E

    F. Bertacco and E. Gwynne. Liouville brownian motion and quantum cones in dimensiond >2.Preprint, available at arXiv:2501.15936, 2025

    work page arXiv 2025

  20. [20]

    Bertoin.L ´evy processes, volume 121 ofCambridge Tracts in Mathematics

    J. Bertoin.L ´evy processes, volume 121 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996

    work page 1996

  21. [21]

    Bettinelli and G

    J. Bettinelli and G. Miermont. Compact Brownian surfaces I: Brownian disks.Probab. Theory Related Fields, 167(3- 4):555–614, 2017

    work page 2017

  22. [22]

    Cercl ´e

    B. Cercl ´e. Liouville conformal field theory on even-dimensional spheres.J. Math. Phys., 63(1):Paper No. 012301, 25, 2022

    work page 2022

  23. [23]

    Chaumont

    L. Chaumont. Conditionings and path decompositions for L ´evy processes.Stochastic Process. Appl., 64(1):39–54, 1996

    work page 1996

  24. [24]

    Chen and M

    Z.-Q. Chen and M. Fukushima.Symmetric Markov processes, time change, and boundary theory, volume 35 ofLondon Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2012

    work page 2012

  25. [25]

    D. A. Croydon. Heat kernel fluctuations for a resistance form with non-uniform volume growth.Proc. Lond. Math. Soc. (3), 94(3):672–694, 2007

    work page 2007

  26. [26]

    D. A. Croydon. Volume growth and heat kernel estimates for the continuum random tree.Probab. Theory Related Fields, 140(1-2):207–238, 2008

    work page 2008

  27. [27]

    D. A. Croydon. Scaling limit for the random walk on the largest connected component of the critical random graph. Publ. Res. Inst. Math. Sci., 48(2):279–338, 2012

    work page 2012

  28. [28]

    David, A

    F. David, A. Kupiainen, R. Rhodes, and V. Vargas. Liouville quantum gravity on the Riemann sphere.Comm. Math. Phys., 342(3):869–907, 2016

    work page 2016

  29. [29]

    Dello Schiavo, R

    L. Dello Schiavo, R. Herry , E. Kopfer, and K.-T. Sturm. Polyharmonic fields and liouville quantum gravity measures on tori of arbitrary dimension: from discrete to continuous.Preprint, available at arXiv:2302.02963, 2023

    work page arXiv 2023

  30. [30]

    Dello Schiavo, R

    L. Dello Schiavo, R. Herry , E. Kopfer, and K.-T. Sturm. Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension.J. Lond. Math. Soc. (2), 110(5):Paper No. e70003, 80, 2024. TWO-SIDED HEAT KERNEL BOUNDS FOR p 8/3-LIOUVILLE BROWNIAN MOTION 131

    work page 2024

  31. [31]

    J. Ding, J. Dub ´edat, A. Dunlap, and H. Falconet. Tightness of Liouville first passage percolation forγ∈(0,2).Publ. Math. Inst. Hautes ´Etudes Sci., 132:353–403, 2020

    work page 2020

  32. [32]

    J. Ding, E. Gwynne, and Z. Zhuang. Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension.Prerint, available at arXiv:2310.03996, 2023

    work page arXiv 2023

  33. [33]

    J. Ding, O. Zeitouni, and F. Zhang. Heat kernel for Liouville Brownian motion and Liouville graph distance.Comm. Math. Phys., 371(2):561–618, 2019

    work page 2019

  34. [34]

    Dub ´edat, H

    J. Dub ´edat, H. Falconet, E. Gwynne, J. Pfeffer, and X. Sun. Weak LQG metrics and Liouville first passage percolation. Probab. Theory Related Fields, 178(1-2):369–436, 2020

    work page 2020

  35. [35]

    Duplantier, J

    B. Duplantier, J. Miller, and S. Sheffield. Liouville quantum gravity as a mating of trees.Ast ´erisque, page arXiv:1409.7055, Sep 2014. To appear

    work page arXiv 2014

  36. [36]

    Duplantier and S

    B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ.Invent. Math., 185(2):333–393, 2011

    work page 2011

  37. [37]

    Fukushima, Y

    M. Fukushima, Y. Oshima, and M. Takeda.Dirichlet forms and symmetric Markov processes, volume 19 ofde Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, extended edition, 2011

    work page 2011

  38. [38]

    Garban, R

    C. Garban, R. Rhodes, and V. Vargas. On the heat kernel and the Dirichlet form of Liouville Brownian motion. Electron. J. Probab., 19:no. 96, 25, 2014

    work page 2014

  39. [39]

    Garban, R

    C. Garban, R. Rhodes, and V. Vargas. Liouville Brownian motion.Ann. Probab., 44(4):3076–3110, 2016

    work page 2016

  40. [40]

    Grigor’yan and N

    A. Grigor’yan and N. Kajino. Localized upper bounds of heat kernels for diffusions via a multiple Dynkin-Hunt formula.Trans. Amer. Math. Soc., 369(2):1025–1060, 2017

    work page 2017

  41. [41]

    Grigor’yan and A

    A. Grigor’yan and A. Telcs. Two-sided estimates of heat kernels on metric measure spaces.Ann. Probab., 40(3):1212–1284, 2012

    work page 2012

  42. [42]

    E. Gwynne. Random surfaces and Liouville quantum gravity .Notices Amer. Math. Soc., 67(4):484–491, 2020

    work page 2020

  43. [43]

    Gwynne and J

    E. Gwynne and J. Miller. Characterizations of SLE κ forκ∈(4,8)on Liouville quantum gravity.arXiv e-prints, page arXiv:1701.05174, Jan. 2017

    work page arXiv 2017

  44. [44]

    Gwynne and J

    E. Gwynne and J. Miller. Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff- Prokhorov-uniform topology , 2017

    work page 2017

  45. [45]

    Gwynne and J

    E. Gwynne and J. Miller. ChordalSLE 6 explorations of a quantum disk.Electron. J. Probab., 23:Paper No. 66, 24, 2018

    work page 2018

  46. [46]

    Gwynne and J

    E. Gwynne and J. Miller. Metric gluing of Brownian and p 8/3-Liouville quantum gravity surfaces.Ann. Probab., 47(4):2303–2358, 2019

    work page 2019

  47. [47]

    Gwynne and J

    E. Gwynne and J. Miller. Confluence of geodesics in Liouville quantum gravity forγ∈(0,2).Ann. Probab., 48(4):1861–1901, 2020

    work page 1901

  48. [48]

    Gwynne and J

    E. Gwynne and J. Miller. Local metrics of the Gaussian free field.Ann. Inst. Fourier (Grenoble), 70(5):2049–2075, 2020

    work page 2049

  49. [49]

    Gwynne and J

    E. Gwynne and J. Miller. Conformal covariance of the Liouville quantum gravity metric forγ∈(0,2).Ann. Inst. Henri Poincar´e Probab. Stat., 57(2):1016–1031, 2021

    work page 2021

  50. [50]

    Gwynne and J

    E. Gwynne and J. Miller. Existence and uniqueness of the Liouville quantum gravity metric forγ∈(0,2).Invent. Math., 223(1):213–333, 2021

    work page 2021

  51. [51]

    Gwynne, J

    E. Gwynne, J. Miller, and S. Sheffield. The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to p 8/3-Liouville quantum gravity .Comm. Math. Phys., 374(2):735–784, 2020

    work page 2020

  52. [52]

    Gwynne, J

    E. Gwynne, J. Miller, and S. Sheffield. The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity .Ann. Probab., 49(4):1677–1717, 2021

    work page 2021

  53. [53]

    Gwynne, J

    E. Gwynne, J. Miller, and S. Sheffield. An invariance principle for ergodic scale-free random environments.Acta Math., 228(2):303–384, 2022

    work page 2022

  54. [54]

    Hughes and J

    L. Hughes and J. Miller. Equivalence of metric gluing and conformal welding inγ-liouville quantum gravity for γ∈(0,2), 2024

    work page 2024

  55. [55]

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

    work page 1985

  56. [56]

    Karatzas and S

    I. Karatzas and S. E. Shreve.Brownian Motion and Stochastic Calculus, volume 113 ofGraduate Texts in Mathematics. Springer-Verlag, New York, Second edition, 1991

    work page 1991

  57. [57]

    T. Kumagai. Anomalous random walks and diffusions: From fractals to random media.Proceedings of the ICM Seoul 2014, Vol. IV:75–94, 2014

    work page 2014

  58. [58]

    A. E. Kyprianou.Introductory lectures on fluctuations of L ´evy processes with applications. Universitext. Springer- Verlag, Berlin, 2006

    work page 2006

  59. [59]

    Lamperti

    J. Lamperti. Continuous state branching processes.Bull. Amer. Math. Soc., 73:382–386, 1967. 132 SEBASTIAN ANDRES, NAOTAKA KAJINO, KONSTANTINOS KAVVADIAS, AND JASON MILLER

    work page 1967

  60. [60]

    G. F. Lawler.Conformally invariant processes in the plane, volume 114 ofMathematical Surveys and Monographs. American Mathematical Society , Providence, RI, 2005

    work page 2005

  61. [61]

    Le Gall.Spatial branching processes, random snakes and partial differential equations

    J.-F. Le Gall.Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics ETH Z¨urich. Birkh¨auser Verlag, Basel, 1999

    work page 1999

  62. [62]

    J.-F. Le Gall. Geodesics in large planar maps and in the Brownian map.Acta Math., 205(2):287–360, 2010

    work page 2010

  63. [63]

    J.-F. Le Gall. Uniqueness and universality of the Brownian map.Ann. Probab., 41(4):2880–2960, 2013

    work page 2013

  64. [64]

    J.-F. Le Gall. Brownian disks and the Brownian snake.Ann. Inst. Henri Poincar´e Probab. Stat., 55(1):237–313, 2019

    work page 2019

  65. [65]

    J.-F. Le Gall. The volume measure of the Brownian sphere is a Hausdorff measure.Electron. J. Probab., 27:Paper No. 113, 28, 2022

    work page 2022

  66. [66]

    Le Gall and M

    J.-F. Le Gall and M. Weill. Conditioned Brownian trees.Ann. Inst. H. Poincar ´e Probab. Statist., 42(4):455–489, 2006

    work page 2006

  67. [67]

    Maillard, R

    P. Maillard, R. Rhodes, V. Vargas, and O. Zeitouni. Liouville heat kernel: regularity and bounds.Ann. Inst. Henri Poincar´e Probab. Stat., 52(3):1281–1320, 2016

    work page 2016

  68. [68]

    Marckert and A

    J.-F. Marckert and A. Mokkadem. Limit of normalized quadrangulations: the Brownian map.Ann. Probab., 34(6):2144–2202, 2006

    work page 2006

  69. [69]

    Miermont

    G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations.Acta Math., 210(2):319–401, 2013

    work page 2013

  70. [70]

    Miller and S

    J. Miller and S. Sheffield. Imaginary geometry I: interacting SLEs.Probab. Theory Related Fields, 164(3–4):553–705, 2016

    work page 2016

  71. [71]

    Miller and S

    J. Miller and S. Sheffield. Imaginary geometry II: reversibility ofSLE κ(ρ1;ρ 2)forκ∈(0,4).Ann. Probab., 44(3):1647–1722, 2016

    work page 2016

  72. [72]

    Miller and S

    J. Miller and S. Sheffield. Imaginary geometry III: reversibility ofSLE κ forκ∈(4,8).Ann. of Math. (2), 184(2):455–486, 2016

    work page 2016

  73. [73]

    Miller and S

    J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding.Ann. Probab., page arXiv:1605.03563, May 2016. To appear

    work page arXiv 2016

  74. [74]

    Miller and S

    J. Miller and S. Sheffield. Quantum Loewner evolution.Duke Math. J., 165(17):3241–3378, 2016

    work page 2016

  75. [75]

    Miller and S

    J. Miller and S. Sheffield. Liouville quantum gravity spheres as matings of finite-diameter trees.Ann. Inst. H. Poincar´e Probab. Statist., 55(3):1712–1750, 2019

    work page 2019

  76. [76]

    Miller and S

    J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map I: the QLE(8/3,0) metric.Invent. Math., 219(1):75–152, 2020

    work page 2020

  77. [77]

    Miller and S

    J. Miller and S. Sheffield. An axiomatic characterization of the Brownian map.J. ´Ec. polytech. Math., 8:609–731, 2021

    work page 2021

  78. [78]

    Miller and S

    J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map III: the conformal structure is deter- mined.Probab. Theory Related Fields, 179(3–4):1183–1211, April 2021

    work page 2021

  79. [79]

    Miller and H

    J. Miller and H. Wu. Intersections of SLE Paths: the double and cut point dimension of SLE.Probab. Theory Related Fields, 167(1–2):45–105, 2017

    work page 2017

  80. [80]

    M. Murugan. On the length of chains in a metric space.J. Funct. Anal., 279(6):108627, 2020

    work page 2020

Showing first 80 references.