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Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity

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Pith's one-line read An infinite-graph random walk that reaches infinity and reflects back is constructed, and its Aldous–Broder forest is shown to be the free uniform spanning forest.

desk verdict A genuinely new random-walk construction of the free spanning forest, with a clean proof for locally finite graphs and an abstract that overstates that scope. read the letter →

arxiv 2506.18827 v2 pith:RFB75TVN submitted 2025-06-23 math.PR

Ewain Gwynne , Jinwoo Sung This is my paper · ORCID
classification math.PR MSC 60J4531C2005C8160J27
keywords randomwalkreflectedoffinfinityenergy-minimizingharmonicfunctionsfreeuniformspanningforestAldous-BroderalgorithmWilson'sTutteembeddingsupercriticalLiouvillequantumgravitydiscreteGaussianfield
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

The paper constructs a continuous-time random walk on any countably infinite, transient graph that almost surely reaches infinity in finite time and bounces back infinitely often. The walk is canonical: its hitting distribution on every finite set is the unique energy-minimizing harmonic measure, and under a sufficiently large vertex rate function it is unique in law. Because the walk is recurrent, the Aldous–Broder algorithm can be run on a single infinite sample path, and the paper proves the resulting forest is the free uniform spanning forest; Wilson's algorithm gives the same forest only when that forest is a single tree. The same machinery defines a Green's function and a discrete Gaussian free field with free boundary at infinity, and a Tutte embedding for infinite planar maps with infinitely many ends, which the paper conjectures converges to supercritical Liouville quantum gravity.

What carries the argument

The carrying object is the energy-minimizing discrete harmonic function: for a finite set A and boundary data φ, the unique function agreeing with φ on A that minimizes the Dirichlet energy over all of V_G. Its values define the harmonic measure hm^x_A(y)=h_y(x), which Property (vi) makes the hitting law of the reflected walk. These functions define transition probabilities of finite approximating Markov chains on one-neighbourhoods B1G_n; the consistency of these chains on nested sets lets the authors couple all approximations and take a limit to get the continuous-time process. Local finiteness is then used to make the Aldous–Broder parent map well-defined.

What would settle it

Take a transient graph with a nontrivial boundary at infinity, for instance two copies of $Z^{3}$ joined by a single edge, and run the reflected walk with any rate w≥w* from a fixed vertex; record the first vertex of a fixed finite set A hit by the walk over many trials. If the empirical hitting frequencies do not converge to the energy-minimizing harmonic measure hm^z_A(y), then Property (vi) of Theorem 1.5 fails and the construction is not the claimed one.

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Extended reading notes

Core claim

The paper's central claim is that on any countably infinite connected graph with finite vertex conductances and transient simple random walk, there is a rate function w* such that for every w≥w* and every starting vertex there is a unique-in-law continuous-time process X taking values in V_G∪{∞}, which is constant on intervals at vertices, is right-continuous, has exponential holding times with the graph's transition probabilities, satisfies the Markov property, returns to its starting point at arbitrarily large times, and has first hitting distribution of every finite set A equal to the energy-minimizing harmonic measure on A. The process reaches ∞ in finite time and reflects infinitely often, with the 'point at infinity' carrying information about which end was hit. From this process, the Aldous–Broder construction has the law of the c-free spanning forest, and a version of Wilson's algorithm agrees with the c-FSF exactly on the event that the c-FSF is a single tree; the Green's function and a discrete GFF with free boundary at infinity are defined, and a Tutte embedding is introduced for infinite maps with infinitely many ends.

Load-bearing premise

The spanning-forest results require G to be locally finite, because the Aldous–Broder parent map is defined as the vertex visited immediately before a first hitting time, and only local finiteness guarantees that such a vertex exists.

Editorial extensions

If this is right

  • Because the reflected walk is recurrent, the Aldous–Broder algorithm can be run forever on a single sample path, giving a path-by-path way to simulate and study the c-free spanning forest.
  • Wilson's algorithm with the reflected walk produces a spanning forest that agrees with the c-FSF if and only if the c-FSF is connected; on the non-connected event the unconditioned Wilson forest is not the FSF because it can create finite components.
  • The Green's function reflected off infinity satisfies the finite-graph identities: harmonicity, symmetry up to factors of π, and a Kirchhoff-type formula P[{x,y}∈FSF] = c(x,y) G_y(x,x)/π(x).
  • The Tutte embedding extends to infinite, multi-ended planar maps, with convex faces and finite-submap approximation, and the paper conjectures that under this embedding supercritical-LQG random planar maps converge to LQG decorated by CLE4.
  • On these maps the paper predicts sharp phase transitions: the free uniform spanning forest is connected for c>16 and has infinitely many components for c<16, and critical percolation occurs for c<95/4 but not for c>95/4.

Reading between the lines

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

  • A natural next step, not pursued by the paper, is to find a loop erasure that is itself allowed to reflect off infinity; such an object would make Wilson's algorithm produce the c-FSF without conditioning on connectedness.
  • The same energy-minimizing harmonic measure could define reflected processes in continuum settings, where the boundary at infinity is the Martin boundary and the discrete construction suggests a canonical 'reflected Brownian motion' on non-compact spaces that returns from infinity in finite time.
  • Numerically, one could test the predicted c=16 FUSF phase transition by simulating the reflected walk on the explicitly constructed supercritical maps and checking whether the Aldous–Broder forest is connected, since the walk visits every vertex almost surely.
  • The free-boundary GFF from Definition 1.20 may give a natural way to define embeddings of multi-ended maps via level-line geometry, which the paper does not explore.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 3 minor

Summary. The paper introduces a continuous-time process on an infinite transient graph that is intended to reach infinity in finite time and reflect off it. The main theorem (Theorem 1.5) asserts existence and uniqueness of such a process satisfying structural properties and an energy-minimizing harmonic-measure hitting property. The authors then use this process to construct the free uniform spanning forest via the Aldous–Broder algorithm (Theorem 1.14, under local finiteness), to give a Wilson-type construction (Theorem 1.16), to define a Green's function and a discrete Gaussian free field (Section 1.5), to define a Tutte embedding for multi-ended planar maps (Section 1.6), and to pose scaling-limit conjectures for supercritical Liouville quantum gravity (Section 6). The paper is written in a clear style, and the finite-dimensional portions of the argument are coherent, but the central projective-limit construction of the reflected walk has a fundamental flaw that invalidates Theorem 1.5 and everything built on it.

Significance. If the construction worked, the paper would provide a novel and potentially powerful representation of the free uniform spanning forest, a natural definition of a Green's function and Gaussian free field with free boundary conditions at infinity, and a concrete embedding for multi-ended random planar maps, thereby opening a route to conjectures about supercritical LQG. The conjectural framework in Section 6 is stimulating. However, the load-bearing existence theorem (Theorem 1.5) is not established: as constructed, the discrete-time process collapses to ordinary random walk, so the claimed reflection mechanism is absent. Because all later results—the FUSF theorems, the Green's function/GFF definitions, and the Tutte embedding—call on Theorem 1.5, the paper's central claims are currently unsupported.

major comments (1)
  1. [Section 3.4, proof of Theorem 1.5 existence] The claim that there are infinitely many ξ with Y_ξ = z follows from recurrence of each Y^n, but as shown in the previous comment, Y is the ordinary random walk, which on a transient graph does not return to z infinitely often. Hence the proof of Property (v) is invalid. Moreover, Property (vi) is proven only for the finite approximations Y^n (Lemma 3.2); the passage to Y uses the asserted agreement of hitting sequences, which fails because Y does not have the harmonic-measure transitions. Thus the existence half of Theorem 1.5 is not established.
minor comments (3)
  1. [Abstract and Section 1.4] The abstract states that the Aldous–Broder algorithm gives the FUSF on 'an infinite graph' with no local-finiteness qualification, but Theorem 1.14 and Definition 1.13 require local finiteness (Section 1.4 explicitly assumes it). The abstract should either assume local finiteness or state that the FUSF application is proved only in the locally finite case.
  2. [Section 3.3, Eq. (3.26)] The definition of X_t = infinity for t outside the intervals [τ_η, τ_{hatη}) is vacuous as written: since the equivalence classes are in bijection with N and the sum of all T_ξ is infinite, the intervals cover [0,∞). The authors should clarify whether they intend the process to visit infinity at some times; as written, X_t is always in VG.
  3. [Section 1.6 and 5.2] The Tutte embedding is defined using harmonic measure on ∂G, but Definition 1.4 requires a finite set A. The text does not state that ∂G is finite; for a general infinite planar map the boundary may be infinite. Please clarify the standing assumption or extend the definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reflected random walk is constructed to have the harmonic-measure property, and the FUSF results are derived from independent finite approximations and classical Aldous–Broder/Wilson theorems.

full rationale

The core derivation is self-contained. Theorem 1.5 defines the reflected random walk by a list of axiomatic properties, including (vi) that its hitting distribution on finite sets is the energy-minimizing harmonic measure; this property is built into the construction via the finite Markov chains in Section 3.1, whose jumps from the boundary neighborhood are defined by that harmonic measure, and is verified in Lemma 3.2. It is therefore a defining feature of the object, not a fitted or predicted output. The FUSF theorems (Theorems 1.14 and 1.16) are proved by comparing the ordered hitting sequences of the reflected walk with ordinary random walks on finite subgraphs (Lemma 4.1), then applying the classical finite Aldous–Broder and Wilson algorithms and the definition of the c-FSF as the total-variation limit of finite spanning trees. No spanning-forest result is assumed as input, and no parameter is fitted to data and later renamed a prediction. The uniqueness part of Theorem 1.5 is proved within the paper. Self-citations such as [BGS24] appear only in the explicitly conjectural supercritical LQG section, which has no proofs and does not support the main theorems; Proposition 6.5 is imported from prior work but concerns only the conjectural application section. The local-finiteness hypothesis needed for Definition 1.13 and Lemma 3.13 is a scope condition for the Aldous–Broder construction, not a circular step.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The main constructions assume a transient, locally finite (for FUSF) graph; no parameters are fitted to data. The rate function w* is chosen by an existence proof and does not enter the vertex order. The LQG section imports standard background results as axioms.

free parameters (1)
  • w* (minimal rate function) = exists, not explicit
    Lemma 3.5 and Theorem 1.5 require w(x) >= w*(x) to make the process recurrent. The proof chooses rates shell-by-shell so that excursion times are small. No numerical value is needed for the theorems; the vertex-order of the process is independent of w.
assumptions (5)
  • domain assumption G is a countably infinite connected graph with pi(x) < infinity for every vertex x.
    Section 1.1 states this as the standing setup.
  • domain assumption The simple random walk on (G,c) is transient.
    Section 1.3 notes the theorem is nontrivial only in the transient case; in the recurrent case the process is just the usual continuous-time walk.
  • domain assumption G is locally finite for the FUSF theorems.
    Section 1.4 says 'throughout this subsection, we assume that G is locally finite'; used in Lemma 3.13 and loop erasure.
  • domain assumption For the Tutte embedding, the external face boundary trace returns to y and enumerates finitely many distinct vertices y_1,...,y_m.
    Section 1.6; formula (1.14) sums over m boundary vertices, so a finite boundary is implicitly required, though this is not stated for general infinite maps.
  • standard math Background results in LQG and CLE are imported from the literature.
    Section 6 relies on prior theorems (e.g., Proposition 6.9 from Pfeffer, dimension results from GHM20, MSW14).
invented entities (1)
  • Continuous-time random walk reflected off infinity
    purpose: The central constructed object; its hitting distribution equals the energy-minimizing harmonic measure and it is recurrent.
    It is a new mathematical object defined in this paper. It has internal falsifiable handles (e.g., its Aldous-Broder output matches the c-FSF), but no external empirical evidence exists.

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Cite this review

Pith. "Pith review of Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity." pith.science (2026). https://pith.science/paper/RFB75TVN

@misc{pith2026250618827,
  author       = {Pith},
  title        = {Pith review of: Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RFB75TVN}},
  note         = {Machine review of arXiv:2506.18827}
}
abstract

Let $\mathcal G$ be an infinite graph -- not necessarily one-ended -- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\mathcal G$ which reaches $\infty$ in finite time and "reflects off of $\infty$" infinitely many times. We show that the Aldous-Broder algorithm for the random walk reflected off of $\infty$ gives the free uniform spanning forest (FUSF) on $\mathcal G$. Furthermore, Wilson's algorithm for the random walk reflected off of $\infty$ gives the FUSF on $\mathcal G$ on the event that the FUSF is connected, but not in general. We also apply the theory of random walk reflected off of $\infty$ to study random planar maps in the universality class of supercritical Liouville quantum gravity (LQG), equivalently LQG with central charge in $(1,25)$. Such random planar maps are infinite, with uncountably many ends. We define a version of the Tutte embedding for such maps under which they conjecturally converge to LQG. We also make several conjectures regarding the qualitative behavior of stochastic processes on such maps -- including the FUSF and critical percolation.

Figures

Figures reproduced from arXiv: 2506.18827 by the authors.

Figure 1
Figure 1. Left: The subgraph Gn (blue) and the vertex set B1Gn \ Gn (black). We have drawn a planar graph for clarity, but of course Gn is not required to be planar. The vertices of Gn \ B1Gn are contained in the three white regions. Right: Twelve steps of the Markov chain Y n on B1Gn defined in (3.2) and (3.3). This Markov chain evolves as a random walk on Gn until it hits a vertex of B1Gn \ Gn, at which point it jumps to a … view at source ↗
Figure 2
Figure 2. An illustration of the iterative construction in Definition [PITH_FULL_IMAGE:figures/full_fig_p040_2.png] view at source ↗

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Works this paper leans on

67 extracted references · 41 canonical work pages

  1. [1]

    H\"older Regularity and Dimension Bounds for Random Curves

    M. Aizenman and A. Burchard. H\"older regularity and dimension bounds for random curves. Duke Math. J. , 99(3):419--453, 1999, arXiv:math/9801027. 1712629

    work page Pith review arXiv 1999

  2. [2]

    The scaling limit of the volume of loop O(n) quadrangulations

    \'E lie A \" d \'e kon , William Da Silva , and XingJian Hu . The scaling limit of the volume of loop O(n) quadrangulations . ArXiv e-prints , February 2024, arXiv:2402.04827

    work page arXiv 2024

  3. [3]

    Supercritical Liouville quantum gravity and CLE$_4$

    Morris Ang and Ewain Gwynne . Supercritical Liouville quantum gravity and CLE _4 . ArXiv e-prints , August 2023, arXiv:2308.11832

    work page Pith review arXiv 2023

  4. [4]

    David J. Aldous. The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math. , 3(4):450--465, 1990. 1069105

    work page 1990

  5. [5]

    Brownian loops and the central charge of a Liouville random surface

    Morris Ang, Minjae Park, Joshua Pfeffer, and Scott Sheffield. Brownian loops and the central charge of a L iouville random surface. Ann. Probab. , 50(4):1322--1358, 2022, arXiv:2005.11845. 4420421

    work page Pith review arXiv 2022

  6. [6]

    M. F. Atiyah. Elliptic operators, discrete groups and von N eumann algebras. In Colloque `` A nalyse et T opologie'' en l' H onneur de H enri C artan ( O rsay, 1974) , Ast\' e risque, No. 32-33, pages 43--72. Soc. Math. France, Paris, 1976. 420729

    work page 1974

  7. [7]

    Borot , J

    G. Borot , J. Bouttier , and E. Guitter . A recursive approach to the O(n) model on random maps via nested loops . Journal of Physics A Mathematical General , 45(4):045002, February 2012, arXiv:1106.0153

    arXiv 2012

  8. [8]

    Upper semicontinuous functions and the S tone approximation theorem

    Gerald Beer. Upper semicontinuous functions and the S tone approximation theorem. J. Approx. Theory , 34(1):1--11, 1982. 647707

    work page 1982

Show all 67 references
  1. [9]

    The dimension of the SLE curves

    Vincent Beffara. The dimension of the SLE curves. Ann. Probab. , 36(4):1421--1452, 2008, arXiv:math/0211322. 2435854 (2009e:60026)

    2008 arXiv

  2. [10]

    Area measures and branched polymers in supercritical Liouville quantum gravity

    Manan Bhatia , Ewain Gwynne , and Jinwoo Sung . Area measures and branched polymers in supercritical Liouville quantum gravity . ArXiv e-prints , October 2024, arXiv:2410.12693

    2024 arXiv

  3. [11]

    Scaling limits of planar maps under the Smith embedding

    Federico Bertacco , Ewain Gwynne , and Scott Sheffield . Scaling limits of planar maps under the Smith embedding . Ann. Probab. , 53(3):1138--1196, 2025, arXiv:2306.02988

    2025 arXiv

  4. [12]

    Uniform spanning forests

    Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm. Uniform spanning forests. Ann. Probab. , 29(1):1--65, 2001. 1825141

    2001

  5. [13]

    Gaussian free field and Liouville quantum gravity

    Nathana \"e l Berestycki and Ellen Powell . Gaussian free field and Liouville quantum gravity . Cambridge University Press, 2024, arXiv:2404.16642. To appear

    2024 arXiv

  6. [14]

    A. Broder. Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science , pages 442--447, 1989

    1989

  7. [15]

    On unique extension of time changed reflecting B rownian motions

    Zhen-Qing Chen and Masatoshi Fukushima. On unique extension of time changed reflecting B rownian motions. Ann. Inst. Henri Poincar\' e Probab. Stat. , 45(3):864--875, 2009, arXiv:0810.5096. 2548508

    2009 arXiv

  8. [16]

    Symmetric M arkov processes, time change, and boundary theory , volume 35 of London Mathematical Society Monographs Series

    Zhen-Qing Chen and Masatoshi Fukushima. Symmetric M arkov processes, time change, and boundary theory , volume 35 of London Mathematical Society Monographs Series . Princeton University Press, Princeton, NJ, 2012. 2849840

    2012

  9. [17]

    The scaling limit of planar maps with large faces

    Nicolas Curien , Gr \'e gory Miermont , and Armand Riera . The scaling limit of planar maps with large faces . ArXiv e-prints , January 2025, arXiv:2501.18566

    2025 arXiv

  10. [18]

    Federico Camia and Charles M. Newman. SLE 6 and CLE 6 from critical percolation. In Probability, geometry and integrable systems , volume 55 of Math. Sci. Res. Inst. Publ. , pages 103--130. Cambridge Univ. Press, Cambridge, 2008, arXiv:math/0611116. 2407594 (2009h:82032)

    2008 arXiv

  11. [19]

    From random walk trajectories to random interlacements , volume 23 of Ensaios Matem\' a ticos [Mathematical Surveys]

    Ji r \' C ern\' y and Augusto Quadros Teixeira. From random walk trajectories to random interlacements , volume 23 of Ensaios Matem\' a ticos [Mathematical Surveys] . Sociedade Brasileira de Matem\' a tica, Rio de Janeiro, 2012. 3014964

    2012

  12. [20]

    Peeling random planar maps , volume 2335 of Lecture Notes in Mathematics

    Nicolas Curien. Peeling random planar maps , volume 2335 of Lecture Notes in Mathematics . Springer, Cham, 2023. \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XLIX---2019, \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour. [Saint-Flour Probability Summer School]. 4703944

    2023

  13. [21]

    Tightness of L iouville first passage percolation for (0,2)

    Jian Ding, Julien Dub\' e dat, Alexander Dunlap, and Hugo Falconet. Tightness of L iouville first passage percolation for (0,2) . Publ. Math. Inst. Hautes \' E tudes Sci. , 132:353--403, 2020, arXiv:1904.08021. 4179836

    2020 arXiv

  14. [22]

    Tightness of supercritical L iouville first passage percolation

    Jian Ding and Ewain Gwynne. Tightness of supercritical L iouville first passage percolation. J. Eur. Math. Soc. (JEMS) , 25(10):3833--3911, 2023, arXiv:2005.13576. 4634685

    2023 arXiv

  15. [23]

    Uniqueness of the critical and supercritical L iouville quantum gravity metrics

    Jian Ding and Ewain Gwynne. Uniqueness of the critical and supercritical L iouville quantum gravity metrics. Proc. Lond. Math. Soc. (3) , 126(1):216--333, 2023, arXiv:2110.00177. 4535021

    2023 arXiv

  16. [24]

    Regularity and confluence of geodesics for the supercritical L iouville quantum gravity metric

    Jian Ding and Ewain Gwynne. Regularity and confluence of geodesics for the supercritical L iouville quantum gravity metric. Probab. Math. Phys. , 5(1):1--54, 2024, arXiv:2104.06502. 4696081

    2024 arXiv

  17. [25]

    Liouville quantum gravity on the R iemann sphere

    Fran c ois David, Antti Kupiainen, R\'emi Rhodes, and Vincent Vargas. Liouville quantum gravity on the R iemann sphere. Comm. Math. Phys. , 342(3):869--907, 2016, arXiv:1410.7318. 3465434

    2016 arXiv

  18. [26]

    An introduction to random interlacements

    Alexander Drewitz, Bal\' a zs R\' a th, and Art\" e m Sapozhnikov. An introduction to random interlacements . SpringerBriefs in Mathematics. Springer, Cham, 2014. 3308116

    2014

  19. [27]

    Critical G aussian multiplicative chaos: convergence of the derivative martingale

    Bertrand Duplantier, R \'e mi Rhodes, Scott Sheffield, and Vincent Vargas. Critical G aussian multiplicative chaos: convergence of the derivative martingale. Ann. Probab. , 42(5):1769--1808, 2014, arXiv:1206.1671. 3262492

    2014 arXiv

  20. [28]

    Renormalization of critical G aussian multiplicative chaos and KPZ relation

    Bertrand Duplantier, R \'e mi Rhodes, Scott Sheffield, and Vincent Vargas. Renormalization of critical G aussian multiplicative chaos and KPZ relation. Comm. Math. Phys. , 330(1):283--330, 2014, arXiv:1212.0529. 3215583

    2014 arXiv

  21. [29]

    Liouville quantum gravity and KPZ

    Bertrand Duplantier and Scott Sheffield. Liouville quantum gravity and KPZ . Invent. Math. , 185(2):333--393, 2011, arXiv:1206.0212. 2819163 (2012f:81251)

    2011 arXiv

  22. [30]

    A construction of reflecting barrier B rownian motions for bounded domains

    Masatoshi Fukushima. A construction of reflecting barrier B rownian motions for bounded domains. Osaka Math. J. , 4:183--215, 1967. 231444

    1967

  23. [31]

    Dirichlet forms and M arkov processes , volume 23 of North-Holland Mathematical Library

    Masatoshi Fukushima. Dirichlet forms and M arkov processes , volume 23 of North-Holland Mathematical Library . North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. 569058

    1980

  24. [32]

    Mean-field behavior for nearest-neighbor percolation in d>10

    Robert Fitzner and Remco van der Hofstad. Mean-field behavior for nearest-neighbor percolation in d>10 . Electron. J. Probab. , 22:Paper No. 43, 65, 2017, arXiv:1506.07977. 3646069

    2017 arXiv

  25. [33]

    Jerison, and Asaf Nachmias

    Ori Gurel-Gurevich, Daniel C. Jerison, and Asaf Nachmias. The D irichlet problem for orthodiagonal maps. Adv. Math. , 374:107379, 53, 2020, arXiv:1906.01613. 4157579

    2020 arXiv

  26. [34]

    An almost sure KPZ relation for SLE and B rownian motion

    Ewain Gwynne, Nina Holden, and Jason Miller. An almost sure KPZ relation for SLE and B rownian motion. Ann. Probab. , 48(2):527--573, 2020, arXiv:1512.01223. 4089487

    2020 arXiv

  27. [35]

    Liouville quantum gravity with matter central charge in (1, 25): a probabilistic approach

    Ewain Gwynne, Nina Holden, Joshua Pfeffer, and Guillaume Remy. Liouville quantum gravity with matter central charge in (1, 25): a probabilistic approach. Comm. Math. Phys. , 376(2):1573--1625, 2020, arXiv:1903.09111. 4103975

    2020 arXiv

  28. [36]

    Existence and uniqueness of the L iouville quantum gravity metric for (0,2)

    Ewain Gwynne and Jason Miller. Existence and uniqueness of the L iouville quantum gravity metric for (0,2) . Invent. Math. , 223(1):213--333, 2021, arXiv:1905.00383. 4199443

    2021 arXiv

  29. [37]

    The T utte E mbedding of the P oisson-- V oronoi T essellation of the B rownian D isk C onverges to 8/3 - L iouville Q uantum G ravity

    Ewain Gwynne, Jason Miller, and Scott Sheffield. The T utte E mbedding of the P oisson-- V oronoi T essellation of the B rownian D isk C onverges to 8/3 - L iouville Q uantum G ravity. Comm. Math. Phys. , 374(2):735--784, 2020, arXiv:1809.02091. 4072229

    2020 arXiv

  30. [38]

    The T utte embedding of the mated- CRT map converges to L iouville quantum gravity

    Ewain Gwynne, Jason Miller, and Scott Sheffield. The T utte embedding of the mated- CRT map converges to L iouville quantum gravity. Ann. Probab. , 49(4):1677--1717, 2021, arXiv:1705.11161

    2021 arXiv

  31. [39]

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

    Ewain Gwynne and Joshua Pfeffer. Connectivity properties of the adjacency graph of SLE_ bubbles for (4,8) . Ann. Probab. , 48(3):1495--1519, 2020, arXiv:1803.04923

    2020 arXiv

  32. [40]

    Random surfaces and L iouville quantum gravity

    Ewain Gwynne. Random surfaces and L iouville quantum gravity. Notices Amer. Math. Soc. , 67(4):484--491, 2020, arXiv:1908.05573. 4186266

    2020 arXiv

  33. [41]

    Mean-field critical behaviour for percolation in high dimensions

    Takashi Hara and Gordon Slade. Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. , 128(2):333--391, 1990. 1043524

    1990

  34. [42]

    Mean-field behaviour and the lace expansion

    Takashi Hara and Gordon Slade. Mean-field behaviour and the lace expansion. In Probability and phase transition ( C ambridge, 1993) , volume 420 of NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. , pages 87--122. Kluwer Acad. Publ., Dordrecht, 1994. 1283177

    1993

  35. [43]

    Convergence of uniform triangulations under the C ardy embedding

    Nina Holden and Xin Sun. Convergence of uniform triangulations under the C ardy embedding. Acta Math. , 230(1):93--203, 2023, arXiv:1905.13207. 4567714

    2023 arXiv

  36. [44]

    Interlacements and the wired uniform spanning forest

    Tom Hutchcroft. Interlacements and the wired uniform spanning forest. Ann. Probab. , 46(2):1170--1200, 2018, arXiv:1512.08509. 3773383

    2018 arXiv

  37. [45]

    Sur le chaos multiplicatif

    Jean-Pierre Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Qu\'ebec , 9(2):105--150, 1985. 829798 (88h:60099a)

    1985

  38. [46]

    Gaskets of O(2) loop-decorated random planar maps

    Emmanuel Kammerer . Gaskets of O(2) loop-decorated random planar maps. ArXiv e-prints , November 2024, arXiv:2411.05541

    2024

  39. [47]

    Uniqueness and universality of the B rownian map

    Jean-Fran c ois Le Gall . Uniqueness and universality of the B rownian map. Ann. Probab. , 41(4):2880--2960, 2013, arXiv:1105.4842. 3112934

    2013 arXiv

  40. [48]

    Probability on Trees and Networks , volume 42 of Cambridge Series in Statistical and Probabilistic Mathematics

    Russell Lyons and Yuval Peres. Probability on Trees and Networks , volume 42 of Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge University Press, New York, 2016. Available at https://rdlyons.pages.iu.edu/. 3616205

    2016

  41. [49]

    Lawler, Oded Schramm, and Wendelin Werner

    Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Values of B rownian intersection exponents. II . P lane exponents. Acta Math. , 187(2):275--308, 2001, arXiv:math/9911084. 1879851 (2002m:60159b)

    2001 arXiv

  42. [50]

    The B rownian map is the scaling limit of uniform random plane quadrangulations

    Gr \'e gory Miermont. The B rownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. , 210(2):319--401, 2013, arXiv:1104.1606. 3070569

    2013 arXiv

  43. [51]

    Liouville quantum gravity and the B rownian map I : the QLE (8/3,0) metric

    Jason Miller and Scott Sheffield. Liouville quantum gravity and the B rownian map I : the QLE (8/3,0) metric. Invent. Math. , 219(1):75--152, 2020, arXiv:1507.00719. 4050102

    2020 arXiv

  44. [52]

    Liouville quantum gravity and the B rownian map II : G eodesics and continuity of the embedding

    Jason Miller and Scott Sheffield. Liouville quantum gravity and the B rownian map II : G eodesics and continuity of the embedding. Ann. Probab. , 49(6):2732--2829, 2021, arXiv:1605.03563. 4348679

    2021 arXiv

  45. [53]

    Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles

    Jason Miller and Lukas Schoug. Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles. Ann. Inst. Henri Poincar\'e Probab. Stat. , 60(4):2267--2296, 2024, arXiv:2201.01748. 4828844

    2024 arXiv

  46. [54]

    Jason Miller, Nike Sun, and David B. Wilson. The H ausdorff dimension of the CLE gasket. Ann. Probab. , 42(4):1644--1665, 2014, arXiv:1403.6076. 3262488

    2014 arXiv

  47. [55]

    Choosing a spanning tree for the integer lattice uniformly

    Robin Pemantle. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. , 19(4):1559--1574, 1991. 1127715

    1991

  48. [56]

    Weak L iouville quantum gravity metrics with matter central charge c (- ,25)

    Joshua Pfeffer. Weak L iouville quantum gravity metrics with matter central charge c (- ,25) . Probab. Math. Phys. , 5(3):545--608, 2024, arXiv:2104.04020. 4765494

    2024 arXiv

  49. [57]

    A. M. Polyakov. Quantum geometry of bosonic strings. Phys. Lett. B , 103(3):207--210, 1981. 623209 (84h:81093a)

    1981

  50. [58]

    K PZ formula for log-infinitely divisible multifractal random measures

    R \'e mi Rhodes and Vincent Vargas. K PZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat. , 15:358--371, 2011, arXiv:0807.1036. 2870520

    2011 arXiv

  51. [59]

    Exploration trees and conformal loop ensembles

    Scott Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. , 147(1):79--129, 2009, arXiv:math/0609167. 2494457 (2010g:60184)

    2009 arXiv

  52. [60]

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

    Scott Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab. , 44(5):3474--3545, 2016, arXiv:1012.4797. 3551203

    2016 arXiv

  53. [61]

    What is a random surface? In I CM --- I nternational C ongress of M athematicians

    Scott Sheffield. What is a random surface? In I CM --- I nternational C ongress of M athematicians. V ol. 2. P lenary lectures , pages 1202--1258. EMS Press, Berlin, 2023, arXiv:2203.02470. 4680280

    2023 arXiv

  54. [62]

    Critical percolation in the plane: conformal invariance, C ardy's formula, scaling limits

    Stanislav Smirnov. Critical percolation in the plane: conformal invariance, C ardy's formula, scaling limits. C. R. Acad. Sci. Paris S\'er. I Math. , 333(3):239--244, 2001, arXiv:0909.4499. 1851632 (2002f:60193)

    2001 arXiv

  55. [63]

    Vacant set of random interlacements and percolation

    Alain-Sol Sznitman. Vacant set of random interlacements and percolation. Ann. of Math. (2) , 171(3):2039--2087, 2010, arXiv:0704.2560. 2680403

    2010 arXiv

  56. [64]

    Teixeira

    A. Teixeira. Interlacement percolation on transient weighted graphs. Electron. J. Probab. , 14:no. 54, 1604--1628, 2009, arXiv:0907.0316. 2525105

    2009 arXiv

  57. [65]

    W. T. Tutte. How to draw a graph. Proc. London Math. Soc. (3) , 13:743--767, 1963. 158387

    1963

  58. [66]

    Generating random spanning trees more quickly than the cover time

    David Bruce Wilson. Generating random spanning trees more quickly than the cover time. In Proceedings of the T wenty-eighth A nnual ACM S ymposium on the T heory of C omputing ( P hiladelphia, PA , 1996) , pages 296--303, New York, 1996. ACM. 1427525

    1996

  59. [67]

    Loop-erased random walk and P oisson kernel on planar graphs

    Ariel Yadin and Amir Yehudayoff. Loop-erased random walk and P oisson kernel on planar graphs. Ann. Probab. , 39(4):1243--1285, 2011, arXiv:0809.2643. 2857239

    2011 arXiv

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