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Ward identities: a geometric point of view and applications

T0 review · 2 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A Schwarzian boundary integral computes the holomorphic variation of the Liouville anomaly in genus zero, making the Virasoro central term a geometric quantity.

desk verdict The Schwarzian anomaly formula is new and clean; the descendant recursions rest on an unproved analytic continuation from [GKRV21], so a referee should demand that dependency be made explicit. read the letter →

arxiv 2608.11550 v1 pith:W6NHIWPF submitted 2026-08-12 math-ph math.MPmath.PR

Baojun Wu , Shengjing Xu This is my paper · ORCID
classification math-phmath.MPmath.PR MSC 81T4017B6881R10
keywords LiouvilleconformalfieldtheoryWardidentitiesVirasoroalgebraSchwarzianderivativeanomalybootstrapBPZequationsShapovalovform
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

This paper aims to identify the scalar anomaly that appears when one infinitesimatically changes the analytic boundary parametrizations of a Liouville conformal field theory amplitude. The paper claims that in genus zero this scalar term is a boundary integral of the Schwarzian derivative of the inverse boundary coordinate, giving a geometric origin for the Virasoro central term. From that formula it derives local Ward identities on disks, annuli, and pairs of pants, and uses them to obtain finite recursions for descendant matrix coefficients. The recursions recover known structural facts—polynomial factorization of pair-of-pants coefficients, the Shapovalov form as an annular zero-weight limit, and the formal chiral vertex-operator coefficients—and yield a geometric proof of smoothness of bulk correlations plus genus-zero BPZ equations for degenerate insertions. A sympathetic reader should care because the paper converts a numerical anomaly into contour data and then into a bootstrapping mechanism for the whole genus-zero theory.

What carries the argument

The load-bearing mechanism is the pairing between infinitesimal boundary deformations and Virasoro modes through meromorphic vector fields. Given a surface with boundary parametrized by $\zeta_k$, a vector field $v$ with a pole at one boundary component pulls back to a Laurent series $-\sum_n [v_k]_n w^{n+1}\partial_w$, so deforming along $v$ inserts $\sum_n([v_k]_n L_n+\overline{[v_k]_n}\tilde L_n)$ into the amplitude; the holomorphic combination isolates $L_{-n}$. The anomaly term in the same deformation is expressed, by Theorem 1.1, as a contour integral of $v$ against the Schwarzian derivative of the inverse boundary coordinate. The local Ward identity balances these Virasoro insertions against the correction term that moves interior marked points, and the recursion closes because $L_k$ acting on a level-$|\nu|$ descendant vanishes for $k>|\nu|$. This yields finite recursions, with polynomial coefficients in conformal weights, that reduce descendant coefficients to primary coefficients.

What would settle it

Take a three-holed sphere with one boundary parametrized by $w+\varepsilon w^2$ and the other two round; evaluate the left-hand side of Theorem 1.1 by differentiating the Liouville anomaly along the flow of $v$ numerically, and compare with the Schwarzian contour integral on the right. Any mismatch at order $\varepsilon^2$ would falsify the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for a genus-zero admissible surface with $b\ge 1$ incoming boundary components and filling $(\hat\Sigma,\hat g)\simeq(\hat{\mathbb C},\hat g)$, the holomorphic variation of the Liouville anomaly under a meromorphic vector field $v=v(z)\partial_z$ is $$\frac12\left(\partial_t $S_L^{0}$(\Sigma_t^v)-i\partial_t $S_L^{0}$(\$Sigma_t^{{iv}}$)\right)\Big|_{t=0}-\frac1{24}\sum_{k=1}^b [v_k]_0 = \frac{i}{24\pi}\sum_{k=1}^b \oint_{\partial_k\Sigma} v(z)\mathcal{S}_{\$zeta_k^{{-1}}$}(z)\,dz,$$ where $\mathcal{S}_{\zeta_k^{-1}}$ is the Schwarzian derivative of the inverse boundary parametrization and the contours follow the incoming orientation. In plain terms, the holomorphic half of the anomaly is completely recorded by Schwarzian boundary integrals; no interior information is needed in genus zero. The paper then combines this formula with the variational formula for Liouville amplitudes, Weyl covariance, and diffeomorphism covariance to obtain a local Ward identity. Choosing vector fields with a pole at one boundary and holomorphy elsewhere turns the identity into recursions that lower the Virasoro level on one boundary at a time, while the Schwarzian term accounts exactly for the central charge. These recursions are the paper's main working tool, applied to disks, annuli, and pairs of pants.

Load-bearing premise

The argument relies on the analytically continued generalized Liouville amplitudes: it assumes the extension to complex boundary weights and to descendant states exists with enough regularity for the recursions, a continuation taken from prior work rather than reproved here.

Editorial extensions

If this is right

  • In the round-boundary case, every normalized descendant coefficient on a disk, annulus, or pair of pants is computed by a finite number of applications of the differential operators $D_{n,z,\alpha}$, so no integration over moduli is needed.
  • The normalized pair-of-pants coefficient factors as $\omega_{\mathcal P_\psi,\nu}(\Delta_\alpha)\omega_{\mathcal P_\psi,\tilde\nu}(\Delta_\alpha)$, with the holomorphic and anti-holomorphic sectors separated; this factorization follows from the recursion itself rather than from a separate bootstrap argument.
  • The zero-weight limit of the normalized annulus coefficient is the Shapovalov form of the Virasoro Verma module, connecting probabilistic amplitudes to the representation-theoretic inner product.
  • Bulk correlation functions of Liouville CFT are smooth in the insertion points on any closed Riemann surface, with a proof that reduces each derivative to finitely many descendant amplitudes.
  • For a degenerate bulk insertion of weight $\alpha_{r,s}$, the null vector in the Liouville module becomes a differential equation of order at most $rs$ with principal part $\partial_x^{rs}$ in flat coordinates; the level-two case is the standard BPZ equation.

Reading between the lines

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

  • Because the boundary-local Proposition 3.1 isolates the anomaly as a contour term plus an interior $\bar\partial V$ correction, the same computation should extend to higher genus; the choice of projective connection would then enter as an obstruction, making the central-charge term computable on arbitrary compact surfaces.
  • The finite recursions are driven entirely by local data such as pole orders and Schwarzian coefficients, so iterating them over a trinion decomposition may yield a deterministic bootstrap algorithm for higher-point and higher-genus correlation numbers, bypassing modular bootstrap constraints.
  • The zero-weight annular limit identifies a probabilistic amplitude with the Shapovalov form; testing this identification at low levels with explicit Gram matrices would provide a numerical check of both the recursions and the module structure.
  • The smoothness argument uses only a zero-weight insertion and the weighted Ward identity; if the same weighted estimates hold for boundary amplitudes, it should prove smoothness of bulk insertions in boundary Liouville CFT as well.
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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

2 major / 6 minor

Summary. The paper develops a geometric framework for local Ward identities in probabilistic Liouville conformal field theory on surfaces with analytic boundary. Its central computation is the holomorphic variation of the Liouville anomaly under boundary deformations: for a genus-zero admissible surface with b incoming boundaries and a meromorphic vector field v, Theorem 1.1, proved as Theorem 3.1, identifies the combination (1/2)(∂_t S_L^0(Σ_t^v)-i∂_t S_L^0(Σ_t^{iv}))|_0 - (1/24)Σ_k [v_k]_0 with the Schwarzian boundary integral (i/24π)Σ_k ∮ v(z) S_{ζ_k^{-1}}(z) dz. The authors combine this formula with the differentiability theorem of Baverez, Guillarmou, Kupiainen, and Rhodes to derive local Ward identities for disks, annuli, and pairs of pants (Lemma 4.1, Propositions 4.3 and 4.4). These yield finite Virasoro recursions, from which the paper recovers the polynomial factorization of normalized pair-of-pants descendant coefficients (Theorem 4.7), the Shapovalov limit of annulus coefficients (Corollary 4.6), and the formal chiral vertex-operator coefficients (Proposition 4.10 and Corollary 4.11). Section 5 gives a geometric proof of smoothness of bulk correlation functions (Theorem 5.2) and derives genus-zero BPZ equations for degenerate insertions (Proposition 5.4 and Corollary 5.5).

Significance. The anomaly formula is the clearest new contribution; the proof of Theorem 3.1 is explicit, local, and effectively self-contained, and I found no sign or factor error in the Schwarzian contour term. If the result stands, it gives a concrete geometric origin for the Virasoro central term as a boundary Schwarzian integral. The paper also adds value by showing that the local Ward identities provide a uniform derivation of several known bootstrap facts, namely polynomial factorization, the Shapovalov limit, and chiral vertex-operator coefficients, and by offering a geometric proof of smoothness of correlations. The authors are appropriately explicit about what is quoted from GKRV21 and BGKR24. The main reservation is that the analytic continuation of generalized amplitudes to complex boundary weights, and the weighted estimates needed for descendant states, are not proved in the manuscript; the cited papers may cover these points, but the precise statements are not located. Overall the contribution is solid, yet the advertised domain of the Ward-identity applications needs to be tightened.

major comments (2)
  1. [Sec. 4.1-4.2 (Prop. 4.3, Thm. 4.7)] The extension of the Ward identities and of the factorization theorem to the physical boundary Re(alpha)=Q of the Seiberg region is asserted by analytic continuation: Proposition 4.3 says that the identities extend to analytically continued generalized states wherever the corresponding generalized amplitudes are defined, and Theorem 4.7 says that the general case follows from analytic continuation. Neither the continuation domain nor the required weighted estimates are stated or proved in this paper. The recursion for L_{-n}Psi_{alpha,nu} is finite, but it must be iterated through intermediate descendant states on the opposite boundaries, so it requires the continuation to be defined in a fixed neighborhood of Re(alpha)=Q for every descendant that appears in the recursion; Theorem 4.7 additionally requires the domain Re(alpha_j)≤Q with sum Re(alpha_j)>2Q. The cited [GKRV21, Section 11.2] and [BGKR24] may well supply this, but the manuscript does not identify the exact theorem or verify that it covers descendant insertions on this domain. Since the factorization (1.2), the Shapovalov limit (4.18), and the chiral vertex identification (4.32) all depend on this step, the proof is incomplete as written. The authors should add a precise analytic-continuation lemma and check that it applies to all descendant states appearing in the recursions.
  2. [Sec. 5.1 (Lemma 5.1, Thm. 5.2)] Lemma 5.1 is the key analytic input for the new geometric proof of smoothness of correlation functions. Its proof consists almost entirely of citations to [GKRV21, Theorem 4.4, Propositions 6.4 and 11.13] and [BGKR24, Lemma 4.10], and the differentiability assertion in equation (5.1) is asserted rather than demonstrated. Because Theorem 5.2 is advertised as an application of the Ward-identity method, the authors should either provide the weighted-estimate argument in detail or state explicitly that Theorem 5.2 is conditional on the collected estimates from the cited papers.
minor comments (6)
  1. [Introduction and Sec. 3] The introduction refers to a boundary-local version as Proposition 3.1, but Section 3 contains only Theorem 3.1; the cross-reference should be corrected.
  2. [Footnote 1, Sec. 4] The footnote says Proposition 4.1 was stated in [GKR24, Proposition 6.6], but the statement proved in the text is Lemma 4.1; the labels should be harmonized.
  3. [Theorem 4.7] The phrase "tripe Young diagrams" is a typo for "triple Young diagrams," and the role of the points x_j=psi_j(0) should be stated explicitly, since the statement of the theorem is otherwise ambiguous.
  4. [Lemma 5.1] The word "implicityly" should read "implicitly."
  5. [Prop. 4.10] The notation Q_{N≥0} for the formal completion of V_{alpha_3} is confusing; it is presumably the direct product over levels, and should be typeset as a product symbol.
  6. [Thm. 5.2 proof] In the definition of E(O), the summation index M is not specified; it should be a finite sum over the finitely many generators U_a in V_fin^0.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructed circularity: the anomaly variation is a direct computation and the Ward identities and recovered factorization rest on independent cited inputs; the only noted issue is a minor self-citation and an inherited analytic-continuation assumption.

full rationale

I found no step in which a claimed prediction reduces by construction to its own input. Theorem 1.1 is proved by differentiating the anomaly functional S_L^0 defined in (2.20) along the real and imaginary flows; the computation in Section 3 uses only the conformal-factor variation (3.8), the exterior-derivative identity (3.11), admissibility near the boundary, and the collar form (3.14). It does not assume the Schwarzian formula it derives. Lemma 4.1 combines the cited variational formula [BGKR24, Theorem 1.2] with Weyl covariance and diffeomorphism invariance, and the new content is the explicit evaluation of the anomaly term through Theorem 1.1; for round boundaries the Schwarzian contribution vanishes because the inverse charts are affine, so no term is being imported as the answer. The recursions in Propositions 4.3, 4.4, and Theorem 4.7 reduce descendant coefficients to primary coefficients without fitting any parameter. The factorization (1.2), the Shapovalov limit, and the chiral vertex-operator coefficients are explicitly recovered, and the paper states that the factorization itself was already proved in [GKRV21]; they are not presented as independent predictions. The main self-citation is [BW26], which supplies the Virasoro module and singular-vector structure used in the BPZ application; this is a separate representation-theoretic input rather than the BPZ equation itself, so it is independent support and not a circular reduction. The analytic-continuation hypothesis quoted from [GKRV21, Section 11.2] is a genuine scope limitation for Theorem 4.7 and Lemma 5.1, but it is an inherited condition on the domain of generalized amplitudes, not a substitution of the conclusion into the input. The score of 2 reflects the presence of a self-citation and the unproved continuation dependence; it does not reflect a constructed circularity.

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

No new fitted constants are introduced. The parameters Q, c_L, Δ_α, and μ are fixed inputs of Liouville theory, not fitted in this paper. The central computation is a derivation from the anomaly functional, and the main external inputs are established probabilistic-CFT theorems and a standard Riemann-Roch fact.

assumptions (5)
  • domain assumption The probabilistic Liouville amplitude formalism, including sewing, Weyl covariance, diffeomorphism invariance, and the boundary-deformation differentiability theorem, is correct.
    Used throughout; cited from [BGKR24, Theorem 1.2] and [GKRV21], not reproven in this paper.
  • domain assumption Generalized descendant states Ψ_{α,ν,ν̃} form an analytic family, and Liouville highest-weight modules have the stated irreducibility and null-vector properties.
    Used for analytic continuation and for the BPZ equations; cited from [BGK+24, Theorem 1.2] and [BW26, Theorems 1.1-1.2].
  • domain assumption The analytic continuation domain of generalized amplitudes covers the complex boundary weights used in Section 4.
    Invoked in the proof of Proposition 4.3 and Theorem 4.7 via [GKRV21, Section 11.2].
  • standard math Riemann-Roch provides meromorphic vector fields with prescribed jets for the smoothness proof.
    Used in Theorem 5.2 to construct the vector field v_x; standard complex geometry.
  • standard math The Liouville anomaly functional satisfies the cocycle identity used in Lemma 4.1.
    This follows from the Dirichlet-energy form of S_L^0 in (2.20); the paper uses it without proof.

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Pith. "Pith review of Ward identities: a geometric point of view and applications." pith.science (2026). https://pith.science/paper/W6NHIWPF

@misc{pith2026260811550,
  author       = {Pith},
  title        = {Pith review of: Ward identities: a geometric point of view and applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W6NHIWPF}},
  note         = {Machine review of arXiv:2608.11550}
}
read the original abstract

The work of Baverez, Guillarmou, Kupiainen, and Rhodes [BGKR24] is the starting point of this paper. It shows that analytic changes of boundary parametrizations act differentiably on Liouville amplitudes, with derivative given by Virasoro operators and a scalar anomaly term. We compute this scalar term. Its holomorphic part is a Schwarzian boundary integral, which gives a geometric explanation of the Virasoro central term. We then derive local Ward identities on disks, annuli, and pairs of pants. They give finite recursions for descendant matrix coefficients. From these recursions we recover the polynomial factorization of normalized pair-of-pants coefficients. In the annular zero-weight limit, we recover the Shapovalov form. For a pair of pants with two incoming boundaries, we recover the formal chiral vertex-operator coefficients. We also give a geometric proof of smoothness in the bulk insertion points and derive the genus-zero arbitrary level BPZ equations for degenerate bulk insertions.

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Works this paper leans on

300 extracted references · 69 canonical work pages

  1. [1]

    The Virasoro structure and the scattering matrix for Liouville conformal field theory , volume=

    Baverez, Guillaume and Guillarmou, Colin and Kupiainen, Antti and Rhodes, Rémi and Vargas, Vincent , year=. The Virasoro structure and the scattering matrix for Liouville conformal field theory , volume=. Probability and Mathematical Physics , publisher=. doi:10.2140/pmp.2024.5.269 , number=

    doi:10.2140/pmp.2024.5.269

  2. [2]

    2024 , eprint=

    Review on the probabilistic construction and Conformal bootstrap in Liouville Theory , author=. 2024 , eprint=

    2024

  3. [3]

    2020 , eprint=

    Higher order BPZ equations for Liouville conformal field theory , author=. 2020 , eprint=

    2020

  4. [4]

    1988 , url=

    The Definition of Conformal Field Theory , author=. 1988 , url=

    1988

  5. [5]

    2024 , eprint=

    Semigroup of annuli in Liouville CFT , author=. 2024 , eprint=

    2024

  6. [6]

    2021 , eprint=

    Segal's axioms and bootstrap for Liouville Theory , author=. 2021 , eprint=

    2021

  7. [7]

    1995 , publisher=

    Algebraic Curves and Riemann Surfaces , author=. 1995 , publisher=

    1995

  8. [8]

    Smoothness of Correlation Functions in Liouville Conformal Field Theory , volume=

    Oikarinen, Joona , year=. Smoothness of Correlation Functions in Liouville Conformal Field Theory , volume=. Annales Henri Poincaré , publisher=. doi:10.1007/s00023-019-00789-0 , number=

    doi:10.1007/s00023-019-00789-0

Show all 300 references
  1. [9]

    Ang, Morris and Holden, Nina and Sun, Xin and Yu, Pu , journal=

  2. [10]

    arXiv preprint arXiv:2211.04580 , year=

    Quantum triangles and imaginary geometry flow lines , author=. arXiv preprint arXiv:2211.04580 , year=

    arXiv

  3. [11]

    Ang, Morris and Sun, Xin and Yu, Pu and Zhuang, Zijie , journal=

  4. [12]

    Liouville quantum gravity weighted by conformal loop ensemble nesting statistics , publisher =

    Holden, Nina and Lehmkuehler, Matthis , keywords =. Liouville quantum gravity weighted by conformal loop ensemble nesting statistics , publisher =. 2022 , copyright =. doi:10.48550/ARXIV.2204.09905 , url =

    2022 doi arXiv

  5. [13]

    , TITLE =

    Fischer, Matthias J. , TITLE =. 2014 , PAGES =. doi:10.1007/978-3-642-45138-6 , URL =

    2014 doi

  6. [14]

    NIST Digital Library of Mathematical Functions

  7. [15]

    and Zamolodchikov, Alexander B

    Fateev, V. and Zamolodchikov, Alexander B. and Zamolodchikov, Alexei B. Boundary L iouville field theory. 1. B oundary state and boundary two point function. 2000. arXiv:hep-th/0001012

    2000 arXiv

  8. [16]

    Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator

    Picco, Marco and Santachiara, Raoul and Viti, Jacopo and Delfino, Gesualdo. Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator. Nucl. Phys. B. 2013. doi:10.1016/j.nuclphysb.2013.07.014. arXiv:1304.6511

    2013 arXiv

  9. [17]

    and Simmons, Jacob J

    Ziff, Robert M. and Simmons, Jacob J. H. and Kleban, Peter. Factorization of correlations in two-dimensional percolation on the plane and torus. J. Phys. A. 2011. doi:10.1088/1751-8113/44/6/065002. arXiv:1011.1101

    2011 arXiv

  10. [18]

    and Simmons, J

    Kleban, P. and Simmons, J. J. H. and Ziff, R. M. Anchored Critical Percolation Clusters and 2-D Electrostatics. Phys. Rev. Lett. 2006. doi:10.1103/PhysRevLett.97.115702. arXiv:cond-mat/0605120

    2006 arXiv

  11. [19]

    On three-point connectivity in two-dimensional percolation

    Delfino, Gesualdo and Viti, Jacopo. On three-point connectivity in two-dimensional percolation. J. Phys. A. 2011. doi:10.1088/1751-8113/44/3/032001. arXiv:1009.1314

    2011 arXiv

  12. [20]

    Kostov, I. K. and Petkova, V. B. Non-rational 2-D quantum gravity. I. World sheet CFT. Nucl. Phys. B. 2007. doi:10.1016/j.nuclphysb.2007.02.014. arXiv:hep-th/0512346

    2007 arXiv

  13. [21]

    Quantum Gravity from Timelike Liouville theory

    Bautista, Teresa and Dabholkar, Atish and Erbin, Harold. Quantum Gravity from Timelike Liouville theory. JHEP. 2019. doi:10.1007/JHEP10(2019)284. arXiv:1905.12689

    2019 arXiv

  14. [22]

    Liouville theory with a central charge less than one

    Ribault, Sylvain and Santachiara, Raoul. Liouville theory with a central charge less than one. JHEP. 2015. doi:10.1007/JHEP08(2015)109. arXiv:1503.02067

    2015 arXiv

  15. [23]

    Analytic Continuation of Liouville Theory

    Harlow, Daniel and Maltz, Jonathan and Witten, Edward. Analytic Continuation of Liouville Theory. JHEP. 2011. doi:10.1007/JHEP12(2011)071. arXiv:1108.4417

    2011 arXiv

  16. [24]

    On the Liouville three point function

    Teschner, Jorg. On the Liouville three point function. Phys. Lett. B. 1995. doi:10.1016/0370-2693(95)01200-A. arXiv:hep-th/9507109

    1995 arXiv

  17. [25]

    Journal of the European Mathematical Society , pages=

    FZZ formula of boundary Liouville CFT via conformal welding , author=. Journal of the European Mathematical Society , pages=

  18. [26]

    Integrability of

    Ang, Morris and Holden, Nina and Sun, Xin , journal =. Integrability of. 2021

    2021

  19. [27]

    Two-valued local sets of the 2

    Aru, Juhan and Sep\'. Two-valued local sets of the 2. Electron. J. Probab. , FJOURNAL =. 2018 , PAGES =. doi:10.1214/18-EJP182 , URL =

    2018 doi

  20. [28]

    Zhan, Dapeng , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00440-018-0895-0 , URL =

    2019 doi

  21. [29]

    and Sheffield, Scott , TITLE =

    Lawler, Gregory F. and Sheffield, Scott , TITLE =. Ann. Probab. , FJOURNAL =. 2011 , NUMBER =. doi:10.1214/10-AOP560 , URL =

    2011 doi

  22. [30]

    Journal of the European Mathematical Society , eprint =

    Non-simple SLE curves are not determined by their range. Journal of the European Mathematical Society , eprint =

  23. [31]

    Probability Theory and Related Fields , pages=

    Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces. Probability Theory and Related Fields , pages=. 2021 , publisher=

    2021

  24. [32]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Universality for SLE(4). ArXiv e-prints , archivePrefix = "arXiv", eprint =

  25. [33]

    Sheffield, Scott , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2007 , NUMBER =

    2007

  26. [34]

    Acta Math

    Schramm, Oded and Sheffield, Scott , TITLE =. Acta Math. , FJOURNAL =. 2009 , NUMBER =. doi:10.1007/s11511-009-0034-y , URL =

    2009 doi

  27. [35]

    Schramm, Oded and Sheffield, Scott , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00440-012-0449-9 , URL =

    2013 doi

  28. [36]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Level Lines of Gaussian Free Field I: Zero-Boundary GFF. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  29. [37]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Level Lines of Gaussian Free Field II: Whole-Plane GFF. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  30. [38]

    Scott Sheffield , year=2005, title=

    2005

  31. [39]

    Miller, Jason and Sheffield, Scott , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s00440-016-0698-0 , URL =

    2016 doi

  32. [40]

    Miller, Jason and Sheffield, Scott , TITLE =. Ann. Probab. , FJOURNAL =. 2016 , NUMBER =. doi:10.1214/14-AOP943 , URL =

    2016 doi

  33. [41]

    Miller, Jason and Sheffield, Scott , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2016 , NUMBER =. doi:10.4007/annals.2016.184.2.3 , URL =

    2016 doi

  34. [42]

    Miller, Jason and Sheffield, Scott , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2017 , NUMBER =

    2017

  35. [43]

    Hu, Xiaoyu and Miller, Jason and Peres, Yuval , TITLE =. Ann. Probab. , FJOURNAL =. 2010 , NUMBER =. doi:10.1214/09-AOP498 , URL =

    2010 doi

  36. [44]

    Commutation relations for

    Dub. Commutation relations for. Comm. Pure Appl. Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1002/cpa.20191 , URL =

    2007 doi

  37. [45]

    Duality of

    Dub. Duality of. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , FJOURNAL =. 2009 , NUMBER =

    2009

  38. [46]

    Dub. S. J. Amer. Math. Soc. , FJOURNAL =. 2009 , NUMBER =. doi:10.1090/S0894-0347-09-00636-5 , URL =

    2009 doi

  39. [47]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    SLE and Virasoro representations: localization. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  40. [48]

    Miller, Jason and Wu, Hao , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00440-015-0677-x , URL =

    2017 doi

  41. [49]

    Miller, Jason , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00220-018-3109-9 , URL =

    2018 doi

  42. [50]

    Duke Math

    Gwynne, Ewain and Miller, Jason and Sun, Xin , TITLE =. Duke Math. J. , FJOURNAL =. 2018 , NUMBER =. doi:10.1215/00127094-2017-0049 , URL =

    2018 doi

  43. [51]

    Electron

    Acosta, Javier , TITLE =. Electron. J. Probab. , FJOURNAL =. 2014 , PAGES =. doi:10.1214/EJP.v19-3170 , URL =

    2014 doi

  44. [52]

    Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields

  45. [53]

    Electron

    Ding, Jian and Zeitouni, Ofer and Zhang, Fuxi , TITLE =. Electron. J. Probab. , FJOURNAL =. 2018 , PAGES =. doi:10.1214/18-EJP189 , URL =

    2018 doi

  46. [54]

    Liouville first passage percolation: geodesic length exponent is strictly larger than 1 at high temperatures

  47. [55]

    Electron

    Ding, Jian and Goswami, Subhajit , TITLE =. Electron. Commun. Probab. , FJOURNAL =. 2017 , PAGES =. doi:10.1214/17-ECP102 , URL =

    2017 doi

  48. [56]

    Upper bounds on Liouville first passage percolation and Watabiki's prediction

  49. [57]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Liouville first passage percolation: the weight exponent is strictly less than 1 at high temperatures. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  50. [58]

    Surveys in stochastic processes , SERIES =

    Slade, Gordon , TITLE =. Surveys in stochastic processes , SERIES =. 2011 , MRCLASS =. doi:10.4171/072-1/9 , URL =

    2011 doi

  51. [59]

    Hara, Takashi and Slade, Gordon , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1992 , NUMBER =

    1992

  52. [60]

    Probability and statistical physics in two and more dimensions , SERIES =

    Bauerschmidt, Roland and Duminil-Copin, Hugo and Goodman, Jesse and Slade, Gordon , TITLE =. Probability and statistical physics in two and more dimensions , SERIES =. 2012 , MRCLASS =

    2012

  53. [61]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Renormalisation group analysis of 4D spin models and self-avoiding walk. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  54. [62]

    Monte Carlo Tests of Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk

    Kennedy, Tom , journal =. Monte Carlo Tests of Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk. 2002 , month =. doi:10.1103/PhysRevLett.88.130601 , url =

    2002 doi

  55. [63]

    Duminil-Copin, Hugo and Smirnov, Stanislav , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2012 , NUMBER =. doi:10.4007/annals.2012.175.3.14 , URL =

    2012 doi

  56. [64]

    Lacoin, Hubert , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00440-013-0520-1 , URL =

    2014 doi

  57. [65]

    Duminil-Copin, Hugo and Hammond, Alan , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00220-013-1811-1 , URL =

    2013 doi

  58. [66]

    Brydges, David and Spencer, Thomas , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1985 , NUMBER =

    1985

  59. [67]

    Hammersley, J. M. , TITLE =. Proc. Cambridge Philos. Soc. , VOLUME =. 1957 , PAGES =

    1957

  60. [68]

    Kesten, Harry , TITLE =. J. Mathematical Phys. , FJOURNAL =. 1964 , PAGES =

    1964

  61. [69]

    Kesten, Harry , TITLE =. J. Mathematical Phys. , FJOURNAL =. 1963 , PAGES =

    1963

  62. [70]

    and Holroyd, Alexander E

    Grimmett, Geoffrey R. and Holroyd, Alexander E. and Peres, Yuval , TITLE =. Ann. Inst. Henri Poincar\'e D , FJOURNAL =. 2014 , NUMBER =. doi:10.4171/AIHPD/3 , URL =

    2014 doi

  63. [71]

    Improved upper bounds for self-avoiding walks in

    P. Improved upper bounds for self-avoiding walks in. Electron. J. Combin. , FJOURNAL =. 2000 , PAGES =

    2000

  64. [72]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Counting self-avoiding walks. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  65. [73]

    1993 , PAGES =

    Madras, Neal and Slade, Gordon , TITLE =. 1993 , PAGES =

    1993

  66. [74]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Self-avoiding walks and connective constants. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  67. [75]

    Conformal Spectra of Polymers on a Random Surface , author =. Phys. Rev. Lett. , volume =. 1988 , month =. doi:10.1103/PhysRevLett.61.1433 , url =

    1988 doi

  68. [76]

    Geometrical critcal phenomena on a random surface of arbitrary genus , author =. Nucl. Phys.B , volume =

  69. [77]

    Wu, Hao , TITLE =. J. Stat. Phys. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s10955-018-1983-3 , URL =

    2018 doi

  70. [78]

    Wu, Hao , TITLE =. Ann. Probab. , FJOURNAL =. 2018 , NUMBER =. doi:10.1214/17-AOP1241 , URL =

    2018 doi

  71. [79]

    Electron

    Wu, Hao and Zhan, Dapeng , TITLE =. Electron. J. Probab. , FJOURNAL =. 2017 , PAGES =. doi:10.1214/17-EJP110 , URL =

    2017 doi

  72. [80]

    ArXiv e-prints , keywords =

    A multifractal boundary spectrum for SLE _ ( ). ArXiv e-prints , keywords =. 2019

    2019

  73. [81]

    , TITLE =

    Kennedy, Tom and Lawler, Gregory F. , TITLE =. Fractal geometry and dynamical systems in pure and applied mathematics. 2013 , MRCLASS =. doi:10.1090/conm/601/11958 , URL =

    2013 doi

  74. [82]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Uniqueness of the welding problem for SLE and Liouville quantum gravity. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  75. [83]

    Yadin, Ariel and Yehudayoff, Amir , TITLE =. Ann. Probab. , FJOURNAL =. 2011 , NUMBER =. doi:10.1214/10-AOP579 , URL =

    2011 doi

  76. [84]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Natural parametrization of percolation interface and pivotal points. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  77. [85]

    Connectivity properties of the adjacency graph of SLE _ bubbles for (4,8)

  78. [86]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Optimal H ``older Continuity and Dimension Properties for SLE with Minkowski Content Parametrization. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  79. [87]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Convergence of loop-erased random walk in the natural parametrization. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  80. [88]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Convergence of radial loop-erased random walk in the natural parametrization. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  81. [89]

    , TITLE =

    Lawler, Gregory F. , TITLE =. J. Math. Soc. Japan , FJOURNAL =. 2015 , NUMBER =. doi:10.2969/jmsj/06741631 , URL =

    2015

  82. [90]

    , TITLE =

    Schramm, Oded and Wilson, David B. , TITLE =. New York J. Math. , FJOURNAL =. 2005 , PAGES =

    2005

  83. [91]

    Israel J

    Schramm, Oded , TITLE =. Israel J. Math. , FJOURNAL =. 2000 , PAGES =. doi:10.1007/BF02803524 , URL =

    2000 doi

  84. [92]

    Rohde, Steffen and Schramm, Oded , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2005 , NUMBER =. doi:10.4007/annals.2005.161.883 , URL =

    2005 doi

  85. [93]

    International

    Schramm, Oded , TITLE =. International. 2007 , MRCLASS =. doi:10.4171/022-1/20 , URL =

    2007 doi

  86. [94]

    Schramm, Oded and Sheffield, Scott , TITLE =. Ann. Probab. , FJOURNAL =. 2005 , NUMBER =. doi:10.1214/009117905000000477 , URL =

    2005 doi

  87. [95]

    and Schramm, Oded and Werner, Wendelin , TITLE =

    Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin , TITLE =. Fractal geometry and applications: a jubilee of. 2004 , MRCLASS =

    2004

  88. [96]

    and Schramm, Oded and Werner, Wendelin , TITLE =

    Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin , TITLE =. Ann. Probab. , FJOURNAL =. 2004 , NUMBER =. doi:10.1214/aop/1079021469 , URL =

    2004

  89. [97]

    Lawler, Gregory and Schramm, Oded and Werner, Wendelin , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2003 , NUMBER =. doi:10.1090/S0894-0347-03-00430-2 , URL =

    2003 doi

  90. [98]

    and Schramm, Oded and Werner, Wendelin , TITLE =

    Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin , TITLE =. Acta Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/BF02392618 , URL =

    2001 doi

  91. [99]

    and Schramm, Oded and Werner, Wendelin , TITLE =

    Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin , TITLE =. Acta Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/BF02392619 , URL =

    2001 doi

  92. [100]

    and Schramm, Oded and Werner, Wendelin , TITLE =

    Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin , TITLE =. Ann. Inst. H. Poincar\'e Probab. Statist. , FJOURNAL =. 2002 , NUMBER =. doi:10.1016/S0246-0203(01)01089-5 , URL =

    2002 doi

  93. [101]

    , TITLE =

    Lawler, G. , TITLE =. Electron. Comm. Probab. , FJOURNAL =. 1996 , PAGES =. doi:10.1214/ECP.v1-975 , URL =

    1996 doi

  94. [102]

    and Schramm, Oded and Werner, Wendelin , TITLE =

    Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin , TITLE =. Math. Res. Lett. , FJOURNAL =. 2001 , NUMBER =. doi:10.4310/MRL.2001.v8.n4.a1 , URL =

    2001 doi

  95. [103]

    , TITLE =

    Johansson Viklund, Fredrik and Lawler, Gregory F. , TITLE =. Acta Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s11511-012-0087-1 , URL =

    2012 doi

  96. [104]

    , TITLE =

    Johansson Viklund, Fredrik and Lawler, Gregory F. , TITLE =. Duke Math. J. , FJOURNAL =. 2011 , NUMBER =. doi:10.1215/00127094-1433376 , URL =

    2011 doi

  97. [105]

    , TITLE =

    Lawler, Gregory F. , TITLE =. Probability and statistical physics in two and more dimensions , SERIES =. 2012 , MRCLASS =

    2012

  98. [106]

    , TITLE =

    Lawler, Gregory F. , TITLE =. Fractal geometry and stochastics. 2009 , MRCLASS =. doi:10.1007/978-3-0346-0030-9_3 , URL =

    2009 doi

  99. [107]

    and Werness, Brent M

    Lawler, Gregory F. and Werness, Brent M. , TITLE =. Ann. Probab. , FJOURNAL =. 2013 , NUMBER =. doi:10.1214/11-AOP695 , URL =

    2013 doi

  100. [108]

    and Lawler, Gregory F

    Alberts, Tom and Kozdron, Michael J. and Lawler, Gregory F. , TITLE =. J. Phys. A , FJOURNAL =. 2012 , NUMBER =. doi:10.1088/1751-8113/45/49/494015 , URL =

    2012 doi

  101. [109]

    and Zhou, Wang , TITLE =

    Lawler, Gregory F. and Zhou, Wang , TITLE =. Ann. Probab. , FJOURNAL =. 2013 , NUMBER =. doi:10.1214/12-AOP742 , URL =

    2013 doi

  102. [110]

    and Masson, Robert , TITLE =

    Alberts, Tom and Kozdron, Michael J. and Masson, Robert , TITLE =. J. Stat. Phys. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s10955-013-0816-7 , URL =

    2013 doi

  103. [111]

    , TITLE =

    Lawler, Gregory F. , TITLE =. Duke Math. J. , FJOURNAL =. 1980 , NUMBER =

    1980

  104. [112]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Basic properties of the natural parametrization for the Schramm-Loewner evolution. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  105. [113]

    , TITLE =

    Lawler, Gregory F. , TITLE =. In the tradition of. 2013 , MRCLASS =. doi:10.1090/conm/590/11726 , URL =

    2013 doi

  106. [114]

    and Lawler, Gregory F

    Field, Laurence S. and Lawler, Gregory F. , TITLE =. J. Stat. Phys. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s10955-013-0729-5 , URL =

    2013 doi

  107. [115]

    Beffara, Vincent , TITLE =. Ann. Probab. , FJOURNAL =. 2008 , NUMBER =. doi:10.1214/07-AOP364 , URL =

    2008 doi

  108. [116]

    , TITLE =

    Lind, Joan R. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2008 , NUMBER =. doi:10.1090/S0002-9947-08-04327-4 , URL =

    2008 doi

  109. [117]

    Smirnov, Stanislav , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2010 , NUMBER =. doi:10.4007/annals.2010.172.1441 , URL =

    2010 doi

  110. [118]

    Smirnov, Stanislav , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1016/S0764-4442(01)01991-7 , URL =

    2001 doi

  111. [119]

    Kemppainen, Antti and Smirnov, Stanislav , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00220-019-03437-0 , URL =

    2019 doi

  112. [120]

    Kemppainen and S

    A. Kemppainen and S. Smirnov , title=. arXiv preprint arXiv:1609.08527 , year=

    arXiv

  113. [121]

    and Smirnov, S

    Beliaev, D. and Smirnov, S. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2009 , NUMBER =. doi:10.1007/s00220-009-0864-7 , URL =

    2009 doi

  114. [122]

    Lectures on probability theory and statistics , SERIES =

    Werner, Wendelin , TITLE =. Lectures on probability theory and statistics , SERIES =. 2004 , MRCLASS =. doi:10.1007/978-3-540-39982-7_2 , URL =

    2004 doi

  115. [123]

    Probability and statistical physics in two and more dimensions , SERIES =

    Duminil-Copin, Hugo and Smirnov, Stanislav , TITLE =. Probability and statistical physics in two and more dimensions , SERIES =. 2012 , MRCLASS =

    2012

  116. [124]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    The Coefficient Problem and Multifractality of Whole-Plane SLE and LLE. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  117. [125]

    Loutsenko, Igor and Yermolayeva, Oksana , TITLE =. J. Stat. Mech. Theory Exp. , FJOURNAL =. 2013 , NUMBER =

    2013

  118. [126]

    Israel J

    Garban, Christophe and Rohde, Steffen and Schramm, Oded , TITLE =. Israel J. Math. , FJOURNAL =. 2012 , PAGES =. doi:10.1007/s11856-011-0161-y , URL =

    2012 doi

  119. [127]

    Zhan, Dapeng , TITLE =. Ann. Probab. , FJOURNAL =. 2008 , NUMBER =. doi:10.1214/07-AOP366 , URL =

    2008 doi

  120. [128]

    Zhan, Dapeng , TITLE =. Ann. Inst. Henri Poincar\'e Probab. Stat. , FJOURNAL =. 2010 , NUMBER =. doi:10.1214/09-AIHP340 , URL =

    2010 doi

  121. [129]

    Zhan, Dapeng , TITLE =. Invent. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s00222-008-0132-z , URL =

    2008 doi

  122. [130]

    Kang, Nam-Gyu , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2007 , NUMBER =. doi:10.1090/S0894-0347-06-00547-9 , URL =

    2007 doi

  123. [131]

    Alberts, Tom and Sheffield, Scott , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2011 , NUMBER =. doi:10.1007/s00440-009-0252-4 , URL =

    2011 doi

  124. [132]

    Electron

    Alberts, Tom and Sheffield, Scott , TITLE =. Electron. J. Probab. , FJOURNAL =. 2008 , PAGES =. doi:10.1214/EJP.v13-515 , URL =

    2008 doi

  125. [133]

    and Rezaei, Mohammad A

    Lawler, Gregory F. and Rezaei, Mohammad A. , TITLE =. Ann. Probab. , FJOURNAL =. 2015 , NUMBER =

    2015

  126. [134]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Hausdorff measure of SLE curves. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  127. [135]

    Alberts, Tom and Binder, Ilia and Viklund, Fredrik , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s00220-016-2587-x , URL =

    2016 doi

  128. [136]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Logarithmic Coefficients and Multifractality of Whole-Plane SLE. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  129. [137]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Remarks on the intersection of SLE _ ( ) curve with the real line. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  130. [138]

    ArXiv e-prints , keywords =

    Liouville Conformal Field Theory on the higher-dimensional sphere. ArXiv e-prints , keywords =. 2019

    2019

  131. [139]

    ArXiv e-prints , keywords =

    Unit boundary length quantum disk: a study of two different perspectives and their equivalence. ArXiv e-prints , keywords =. 2019

    2019

  132. [140]

    ArXiv e-prints , keywords =

    Mating of trees for random planar maps and Liouville quantum gravity: a survey. ArXiv e-prints , keywords =. 2019

    2019

  133. [141]

    The tail expansion of

    Rhodes, R\'. The tail expansion of. Ann. Probab. , FJOURNAL =. 2019 , NUMBER =. doi:10.1214/18-AOP1333 , URL =

    2019 doi

  134. [142]

    ArXiv e-prints , keywords =

    Universal tail profile of Gaussian multiplicative chaos. ArXiv e-prints , keywords =. 2019

    2019

  135. [143]

    Peyri\`ere, Jacques , TITLE =. C. R. Acad. Sci. Paris S\'. 1974 , PAGES =

    1974

  136. [144]

    A general class of quantum fields without cut-offs in two space-time dimensions , JOURNAL =

    H. A general class of quantum fields without cut-offs in two space-time dimensions , JOURNAL =. 1971 , PAGES =

    1971

  137. [145]

    ArXiv e-prints , keywords =

    Gaussian multiplicative chaos through the lens of the 2D Gaussian free field. ArXiv e-prints , keywords =. 2017

    2017

  138. [146]

    ArXiv e-prints , keywords =

    Lecture notes on Liouville theory and the DOZZ formula. ArXiv e-prints , keywords =. 2017

    2017

  139. [147]

    arXiv e-prints , keywords =

    The dimension of the boundary of a Liouville quantum gravity metric ball. arXiv e-prints , keywords =. 2019

    2019

  140. [148]

    Notices Amer

    Gwynne, Ewain , TITLE =. Notices Amer. Math. Soc. , FJOURNAL =. 2020 , NUMBER =. doi:10.1090/noti , URL =

    2020 doi

  141. [149]

    ArXiv e-prints , keywords =

    Dynamical Liouville. ArXiv e-prints , keywords =. 2018

    2018

  142. [150]

    ArXiv e-prints , keywords =

    Stochastic Ricci Flow on Compact Surfaces. ArXiv e-prints , keywords =. 2019

    2019

  143. [151]

    arXiv e-prints , archivePrefix = "arXiv", eprint =

    Conformal welding for critical Liouville quantum gravity. arXiv e-prints , archivePrefix = "arXiv", eprint =

  144. [152]

    KPZ formulas for the Liouville quantum gravity metric. 2019

    2019

  145. [153]

    ArXiv e-prints , keywords =

    Liouville dynamical percolation. ArXiv e-prints , keywords =. 2019

    2019

  146. [154]

    Gwynne, Ewain and Miller, Jason , TITLE =. Invent. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00222-020-00991-6 , URL =

    2021 doi

  147. [155]

    Gwynne and J

    E. Gwynne and J. Miller , title=

  148. [156]

    Gwynne and J

    E. Gwynne and J. Miller , title=. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  149. [157]

    J. Dub. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  150. [158]

    Ang , title = "

    M. Ang , title = ". ArXiv e-prints , keywords =. 2019

    2019

  151. [159]

    Tightness of

    Ding, Jian and Dub\'. Tightness of. Publ. Math. Inst. Hautes \'. 2020 , PAGES =. doi:10.1007/s10240-020-00121-1 , URL =

    2020 doi

  152. [160]

    Bounds for distances and geodesic dimension in Liouville first passage percolation. 2019

    2019

  153. [161]

    Communications in Mathematical Physics. 2019

    2019

  154. [162]

    2021 , doi =

    Annales de l'Institut Henri Poincar. 2021 , doi =

    2021

  155. [163]

    ArXiv e-prints , keywords =

    Decompositions of log-correlated fields with applications. ArXiv e-prints , keywords =. 2018

    2018

  156. [164]

    ArXiv e-prints , keywords =

    A probabilistic approach of ultraviolet renormalisation in the boundary Sine-Gordon model. ArXiv e-prints , keywords =. 2019

    2019

  157. [165]

    Shamov, Alexander , TITLE =. J. Funct. Anal. , FJOURNAL =. 2016 , NUMBER =. doi:10.1016/j.jfa.2016.03.001 , URL =

    2016 doi

  158. [166]

    ArXiv e-prints , keywords =

    Subsequential scaling limits for Liouville graph distance. ArXiv e-prints , keywords =

  159. [167]

    ArXiv e-prints , keywords =

    The geodesics in Liouville quantum gravity are not Schramm-Loewner evolutions. ArXiv e-prints , keywords =

  160. [168]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Path integral approach to analytic continuation of Liouville theory: the pencil region. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  161. [169]

    Liouville metric of star-scale invariant fields: tails and Weyl scaling

  162. [170]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to 8/3 -Liouville quantum gravity. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  163. [171]

    The fractal dimension of L iouville quantum gravity: universality, monotonicity, and bounds

  164. [172]

    Heat kernel for Liouville Brownian motion and Liouville graph distance

  165. [173]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Liouville quantum gravity as a metric space and a scaling limit. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  166. [174]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Natural parametrization of SLE: the Gaussian free field point of view. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  167. [175]

    The Annals of Probability , volume=

    The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity. The Annals of Probability , volume=. 2021 , publisher=

    2021

  168. [176]

    Holden, Nina and Sun, Xin , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00220-018-3149-1 , URL =

    2018 doi

  169. [177]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Equivalence of Liouville measure and Gaussian free field. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  170. [178]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Field-measure correspondence in Liouville quantum gravity almost surely commutes with all conformal maps simultaneously. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  171. [179]

    A distance exponent for Liouville quantum gravity

  172. [180]

    Dimension transformation formula for conformal maps into the complement of an SLE curve

  173. [181]

    Aru, Juhan , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00440-014-0597-1 , eprint =

    2015 doi

  174. [182]

    Andres, Sebastian and Kajino, Naotaka , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2016 , NUMBER =

    2016

  175. [183]

    and Rhodes, R

    Maillard, P. and Rhodes, R. and Vargas, V. and Zeitouni, O. , TITLE =. Ann. Inst. Henri Poincar\'e Probab. Stat. , FJOURNAL =. 2016 , NUMBER =

    2016

  176. [184]

    Aru, Juhan and Huang, Yichao and Sun, Xin , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2017 , NUMBER =

    2017

  177. [185]

    An almost sure KPZ relation for SLE and Brownian motion

  178. [186]

    Critical

    Duplantier, Bertrand and Rhodes, R. Critical. Ann. Probab. , FJOURNAL =. 2014 , NUMBER =. doi:10.1214/13-AOP890 , URL =

    2014 doi

  179. [187]

    Renormalization of critical

    Duplantier, Bertrand and Rhodes, R. Renormalization of critical. Comm. Math. Phys. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00220-014-2000-6 , URL =

    2014 doi

  180. [188]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  181. [189]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Liouville measure as a multiplicative cascade via level sets of the Gaussian free field. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  182. [190]

    Critical

    Aru, Juhan and Powell, Ellen and Sep\'. Critical. Electron. Commun. Probab. , FJOURNAL =. 2019 , PAGES =. doi:10.1214/19-ECP209 , URL =

    2019 doi

  183. [191]

    Duplantier, Bertrand and Sheffield, Scott , TITLE =. Invent. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1007/s00222-010-0308-1 , URL =

    2011 doi

  184. [192]

    Sheffield, Scott , TITLE =. Ann. Probab. , FJOURNAL =. 2016 , NUMBER =. doi:10.1214/15-AOP1055 , URL =

    2016 doi

  185. [193]

    Sheffield, Scott , TITLE =. Ann. Probab. , FJOURNAL =. 2016 , NUMBER =. doi:10.1214/15-AOP1061 , URL =

    2016 doi

  186. [194]

    Duke Math

    Miller, Jason and Sheffield, Scott , TITLE =. Duke Math. J. , FJOURNAL =. 2016 , NUMBER =. doi:10.1215/00127094-3627096 , URL =

    2016 doi

  187. [195]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Liouville quantum gravity as a mating of trees. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  188. [196]

    Gwynne, Ewain and Holden, Nina and Miller, Jason and Sun, Xin , TITLE =. Ann. Inst. Henri Poincar\'e Probab. Stat. , FJOURNAL =. 2017 , NUMBER =

    2017

  189. [197]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Complex Gaussian multiplicative chaos. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  190. [198]

    Gaussian multiplicative chaos and applications:

    Rhodes, R. Gaussian multiplicative chaos and applications:. Probab. Surv. , FJOURNAL =. 2014 , PAGES =. doi:10.1214/13-PS218 , URL =

    2014 doi

  191. [199]

    Robert, Raoul and Vargas, Vincent , TITLE =. Ann. Probab. , FJOURNAL =. 2010 , NUMBER =. doi:10.1214/09-AOP490 , URL =

    2010 doi

  192. [200]

    Lognormal

    Allez, Romain and Rhodes, R. Lognormal. Probab. Theory Related Fields , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00440-012-0412-9 , URL =

    2013 doi

  193. [201]

    Levy multiplicative chaos and star scale invariant random measures , JOURNAL =

    Rhodes, R. Levy multiplicative chaos and star scale invariant random measures , JOURNAL =. 2014 , NUMBER =. doi:10.1214/12-AOP810 , URL =

    2014 doi

  194. [202]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Liouville heat kernel: regularity and bounds. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  195. [203]

    Molchan, G. M. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1996 , NUMBER =

    1996

  196. [204]

    Garban, Christophe and Rhodes, R\'emi and Vargas, Vincent , TITLE =. Ann. Probab. , FJOURNAL =. 2016 , NUMBER =. doi:10.1214/15-AOP1042 , URL =

    2016 doi

  197. [205]

    Berestycki, Nathana\"el , TITLE =. Ann. Inst. Henri Poincar\'e Probab. Stat. , FJOURNAL =. 2015 , NUMBER =. doi:10.1214/14-AIHP605 , URL =

    2015 doi

  198. [206]

    Electron

    Garban, Christophe and Rhodes, R\'emi and Vargas, Vincent , TITLE =. Electron. J. Probab. , FJOURNAL =. 2014 , PAGES =

    2014

  199. [207]

    Berestycki, Nathana\"el and Garban, Christophe and Rhodes, R\'emi and Vargas, Vincent , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2016 , NUMBER =

    2016

  200. [208]

    Spectral

    Rhodes, R. Spectral. Ann. Henri Poincar\'e , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00023-013-0308-y , URL =

    2014 doi

  201. [209]

    Potential Anal

    Rhodes, R\'emi and Vargas, Vincent , TITLE =. Potential Anal. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s11118-015-9467-4 , URL =

    2015 doi

  202. [210]

    Jackson, Henry , TITLE =. Ann. Inst. Henri Poincar\'. 2018 , NUMBER =. doi:10.1214/16-AIHP803 , URL =

    2018 doi

  203. [211]

    Curien, Nicolas , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00220-014-2196-5 , URL =

    2015 doi

  204. [212]

    Scaling limits for the peeling process on random maps , JOURNAL =

    Curien, Nicolas and Le Gall, Jean-Fran. Scaling limits for the peeling process on random maps , JOURNAL =. 2017 , NUMBER =

    2017

  205. [213]

    , TITLE =

    Angel, O. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00039-003-0436-5 , URL =

    2003 doi

  206. [214]

    Gurel-Gurevich, Ori and Nachmias, Asaf , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2013 , NUMBER =. doi:10.4007/annals.2013.177.2.10 , URL =

    2013 doi

  207. [215]

    and Rohde, Steffen , TITLE =

    Gill, James T. and Rohde, Steffen , TITLE =. Rev. Mat. Iberoam. , FJOURNAL =. 2013 , NUMBER =. doi:10.4171/RMI/749 , URL =

    2013 doi

  208. [216]

    The Annals of Probability , volume=

    Boundaries of planar graphs, via circle packings , author=. The Annals of Probability , volume=. 2016 , publisher=

    2016

  209. [217]

    The boundary of a square tiling of a graph coincides with the

    Georgakopoulos, Agelos , journal=. The boundary of a square tiling of a graph coincides with the. 2016 , publisher=

    2016

  210. [218]

    Inventiones mathematicae , volume=

    Unimodular hyperbolic triangulations: Circle packing and random walk , author=. Inventiones mathematicae , volume=. 2016 , publisher=

    2016

  211. [219]

    Kahane, Jean-Pierre , TITLE =. Ann. Sci. Math. Qu\'ebec , FJOURNAL =. 1985 , NUMBER =

    1985

  212. [220]

    Ast\'erisque , FJOURNAL =

    Garban, Christophe , TITLE =. Ast\'erisque , FJOURNAL =. 2013 , PAGES =

    2013

  213. [221]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Basic properties of critical lognormal multiplicative chaos. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  214. [222]

    Barral, Julien and Kupiainen, Antti and Nikula, Miika and Saksman, Eero and Webb, Christian , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00220-013-1829-4 , URL =

    2014 doi

  215. [223]

    Gaussian multiplicative chaos and

    Barral, Julien and Jin, Xiong and Rhodes, R. Gaussian multiplicative chaos and. Comm. Math. Phys. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00220-013-1769-z , URL =

    2013 doi

  216. [224]

    Benjamini, Itai and Schramm, Oded , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2009 , NUMBER =. doi:10.1007/s00220-009-0752-1 , URL =

    2009 doi

  217. [225]

    Rhodes, R. K. ESAIM Probab. Stat. , FJOURNAL =. 2011 , PAGES =. doi:10.1051/ps/2010007 , URL =

    2011

  218. [226]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Backward SLE and the symmetry of the welding. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  219. [227]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    The hull process of the Brownian plane. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  220. [228]

    Curien, Nicolas and. The. J. Theoret. Probab. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s10959-013-0485-0 , URL =

    2014 doi

  221. [229]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    First-passage percolation and local modifications of distances in random triangulations. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  222. [230]

    Selected works of

    Schramm, Oded and Smirnov, Stanislav , TITLE =. Selected works of. 2011 , MRCLASS =. doi:10.1007/978-1-4419-9675-6_35 , URL =

    2011 doi

  223. [231]

    Miller, Jason and Sheffield, Scott , TITLE =. Ann. Inst. Henri Poincar\'. 2019 , NUMBER =. doi:10.1214/18-aihp932 , URL =

    2019 doi

  224. [232]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    An axiomatic characterization of the Brownian map. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  225. [233]

    Liouville quantum gravity on the unit disk , JOURNAL =

    Huang, Yichao and Rhodes, R\'. Liouville quantum gravity on the unit disk , JOURNAL =. 2018 , NUMBER =. doi:10.1214/17-AIHP852 , URL =

    2018 doi

  226. [234]

    Duke Math

    Remy, Guillaume , TITLE =. Duke Math. J. , FJOURNAL =. 2020 , NUMBER =. doi:10.1215/00127094-2019-0045 , URL =

    2020 doi

  227. [235]

    ArXiv e-prints , keywords =

    The distribution of Gaussian multiplicative chaos on the unit interval. ArXiv e-prints , keywords =

  228. [236]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Local Conformal Structure of Liouville Quantum Gravity. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  229. [237]

    Integrability of Liouville theory: proof of the DOZZ Formula

  230. [238]

    Liouville quantum gravity on the

    David, Fran. Liouville quantum gravity on the. Comm. Math. Phys. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s00220-016-2572-4 , URL =

    2016 doi

  231. [239]

    Liouville quantum gravity on complex tori , JOURNAL =

    David, Fran. Liouville quantum gravity on complex tori , JOURNAL =. 2016 , NUMBER =. doi:10.1063/1.4938107 , URL =

    2016 doi

  232. [240]

    Polyakov's formulation of

    Guillarmou, Colin and Rhodes, R\'. Polyakov's formulation of. Publ. Math. Inst. Hautes \'. 2019 , PAGES =. doi:10.1007/s10240-019-00109-6 , URL =

    2019 doi

  233. [241]

    Remy, Guillaume , TITLE =. J. Math. Phys. , FJOURNAL =. 2018 , NUMBER =. doi:10.1063/1.5030409 , URL =

    2018 doi

  234. [242]

    Electron

    Berestycki, Nathana\"el , TITLE =. Electron. Commun. Probab. , FJOURNAL =. 2017 , PAGES =. doi:10.1214/17-ECP58 , URL =

    2017 doi

  235. [243]

    Inventiones Mathematicae , eprint =

    Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric. Inventiones Mathematicae , eprint =

  236. [244]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  237. [245]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Liouville quantum gravity and the Brownian map III: the conformal structure is determined. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  238. [246]

    International Math Research Notices

    Conformal invariance of CLE \_ on the Riemann sphere for (4,8). International Math Research Notices

  239. [247]

    arXiv e-prints , keywords =

    Extremal distance and conformal radius of a CLE\_4 loop. arXiv e-prints , keywords =

  240. [248]

    Zhan, Dapeng , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00440-020-01011-7 , URL =

    2021 doi

  241. [249]

    Lupu, Titus , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2019 , NUMBER =. doi:10.4171/JEMS/859 , URL =

    2019 doi

  242. [250]

    The scaling limit of critical

    Benoist, St\'. The scaling limit of critical. Ann. Probab. , FJOURNAL =. 2019 , NUMBER =. doi:10.1214/18-AOP1301 , URL =

    2019 doi

  243. [251]

    Forum Math

    Miller, Jason and Sheffield, Scott and Werner, Wendelin , TITLE =. Forum Math. Pi , FJOURNAL =. 2017 , PAGES =

    2017

  244. [252]

    Duke Math

    Sheffield, Scott , TITLE =. Duke Math. J. , FJOURNAL =. 2009 , NUMBER =. doi:10.1215/00127094-2009-007 , URL =

    2009 doi

  245. [253]

    Sheffield, Scott and Werner, Wendelin , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2012 , NUMBER =. doi:10.4007/annals.2012.176.3.8 , URL =

    2012 doi

  246. [254]

    Miller, Jason and Sheffield, Scott , title=

  247. [255]

    and Wu, Hao , title=

    Sheffield, Scott and Watson, Samuel S. and Wu, Hao , title=

  248. [256]

    and Wilson, David B

    Miller, Jason and Watson, Samuel S. and Wilson, David B. , TITLE =. Ann. Probab. , FJOURNAL =. 2016 , NUMBER =. doi:10.1214/14-AOP995 , eprint =

    2016 doi

  249. [257]

    and Wilson, David B

    Miller, Jason and Watson, Samuel S. and Wilson, David B. , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00440-014-0604-6 , URL =

    2015 doi

  250. [258]

    Electron

    Werner, Wendelin and Wu, Hao , TITLE =. Electron. J. Probab. , FJOURNAL =. 2013 , PAGES =. doi:10.1214/EJP.v18-2376 , URL =

    2013 doi

  251. [259]

    Werner, Wendelin and Wu, Hao , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00220-013-1719-9 , URL =

    2013 doi

  252. [260]

    and Werner, Wendelin , TITLE =

    Lawler, Gregory F. and Werner, Wendelin , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2004 , NUMBER =. doi:10.1007/s00440-003-0319-6 , URL =

    2004 doi

  253. [261]

    Kemppainen, Antti and Werner, Wendelin , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s00440-015-0647-3 , URL =

    2016 doi

  254. [262]

    Kenyon, Richard , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00220-013-1881-0 , URL =

    2014 doi

  255. [263]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Double dimers, conformal loop ensembles and isomonodromic deformations. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  256. [264]

    , TITLE =

    Miller, Jason and Sun, Nike and Wilson, David B. , TITLE =. Ann. Probab. , FJOURNAL =. 2014 , NUMBER =. doi:10.1214/12-AOP820 , URL =

    2014 doi

  257. [265]

    Le Jan, Yves , TITLE =. Ann. Probab. , FJOURNAL =. 2010 , NUMBER =. doi:10.1214/09-AOP509 , URL =

    2010 doi

  258. [266]

    2011 , PAGES =

    Le Jan, Yves , TITLE =. 2011 , PAGES =. doi:10.1007/978-3-642-21216-1 , URL =

    2011 doi

  259. [267]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Random walk loop soups and conformal loop ensembles. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  260. [268]

    , TITLE =

    Camia, Federico and Newman, Charles M. , TITLE =. Probability, geometry and integrable systems , SERIES =. 2008 , MRCLASS =

    2008

  261. [269]

    and Wu, Hao , TITLE =

    Sheffield, Scott and Watson, Samuel S. and Wu, Hao , TITLE =. Ann. Inst. Henri Poincar\'. 2017 , NUMBER =. doi:10.1214/15-AIHP726 , URL =

    2017 doi

  262. [270]

    , TITLE =

    Schramm, Oded and Sheffield, Scott and Wilson, David B. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2009 , NUMBER =. doi:10.1007/s00220-009-0731-6 , URL =

    2009 doi

  263. [271]

    Random soups, carpets and fractal dimensions , JOURNAL =

    Nacu,. Random soups, carpets and fractal dimensions , JOURNAL =. 2011 , NUMBER =. doi:10.1112/jlms/jdq094 , URL =

    2011 doi

  264. [272]

    Tutte, W. T. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1963 , PAGES =. doi:10.1112/plms/s3-13.1.743 , URL =

    1963 doi

  265. [273]

    ArXiv e-prints , keywords =

    Scaling limit of large triangulations of polygons. ArXiv e-prints , keywords =. 2019

    2019

  266. [274]

    Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms , series =

    Schnyder, Walter , title =. Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms , series =. 1990 , isbn =

    1990

  267. [275]

    Order , FJOURNAL =

    Schnyder, Walter , TITLE =. Order , FJOURNAL =. 1989 , NUMBER =. doi:10.1007/BF00353652 , URL =

    1989 doi

  268. [276]

    ArXiv e-prints , keywords =

    A combinatorial criterion for macroscopic circles in planar triangulations. ArXiv e-prints , keywords =. 2019

    2019

  269. [277]

    ArXiv e-prints , keywords =

    Iterated foldings of discrete spaces and their limits: candidates for the role of Brownian map in higher dimensions. ArXiv e-prints , keywords =. 2019

    2019

  270. [278]

    2015", month =

    Every planar graph with the Liouville property is amenable. ArXiv e-prints , year = "2015", month = "Feb", archivePrefix =

    2015

  271. [279]

    ArXiv e-prints , keywords =

    Convergence of uniform triangulations under the Cardy embedding. ArXiv e-prints , keywords =. 2019

    2019

  272. [280]

    arXiv e-prints , keywords =

    Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense. arXiv e-prints , keywords =. 2019

    2019

  273. [281]

    Stability of geodesics in the

    Angel, Omer and Kolesnik, Brett and Miermont, Gr\'. Stability of geodesics in the. Ann. Probab. , FJOURNAL =. 2017 , NUMBER =. doi:10.1214/16-AOP1140 , URL =

    2017 doi

  274. [282]

    ArXiv e-prints , keywords =

    External diffusion limited aggregation on a spanning-tree-weighted random planar map. ArXiv e-prints , keywords =

  275. [283]

    ArXiv e-prints , keywords =

    Tree-decorated planar maps. ArXiv e-prints , keywords =

  276. [284]

    ArXiv e-prints , keywords =

    Local convergence of large random triangulations coupled with an Ising model. ArXiv e-prints , keywords =

  277. [285]

    Addario-Berry, Louigi and Leavitt, Nicholas , TITLE =. Ann. Inst. Henri Poincar\'. 2016 , NUMBER =. doi:10.1214/14-AIHP661 , URL =

    2016 doi

  278. [286]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen). ArXiv e-prints , archivePrefix = "arXiv", eprint =

  279. [287]

    Electron

    Miermont, Gr\'egory , TITLE =. Electron. Commun. Probab. , FJOURNAL =. 2008 , PAGES =. doi:10.1214/ECP.v13-1368 , URL =

    2008 doi

  280. [288]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    On tessellations of random maps and the t_g -recurrence. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  281. [289]

    Guitter, Emmanuel , TITLE =. J. Stat. Mech. Theory Exp. , FJOURNAL =. 2017 , NUMBER =

    2017

  282. [290]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    An invariance principle for ergodic scale-free random environments. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  283. [291]

    Electron

    Harmonic functions on mated- CRT maps. Electron. J. Probab. , FJOURNAL =. 2019 , PAGES =

    2019

  284. [292]

    ArXiv Mathematics e-prints , eprint =

    Random walks of circle packings. ArXiv Mathematics e-prints , eprint =

  285. [293]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Percolation on triangulations: a bijective path to Liouville quantum gravity. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  286. [294]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Anomalous diffusion of random walk on random planar maps. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  287. [295]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Hyperbolic and Parabolic Unimodular Random Maps. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  288. [296]

    Electron

    Aldous, David and Lyons, Russell , TITLE =. Electron. J. Probab. , FJOURNAL =. 2007 , PAGES =

    2007

  289. [297]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Random walk on random planar maps: spectral dimension, resistance, and displacement. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  290. [298]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    A mating-of-trees approach for graph distances in random planar maps. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  291. [299]

    Geometric and spectral properties of causal maps

  292. [300]

    Brownian disks and the

    Le Gall, Jean-Fran. Brownian disks and the. Ann. Inst. Henri Poincar\'. 2019 , NUMBER =. doi:10.1214/18-aihp882 , URL =

    2019 doi

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