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The scaling limit of planar maps with large faces

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper constructs, for each α∈(1,2), a universal random compact metric space S_α — the α-stable carpet in the dilute phase and α-stable gasket in the dense phase — and proves that all non-generic critical Boltzmann planar maps of…

desk verdict A very strong paper: explicit stable limits for non-generic Boltzmann maps, with the re-rooting reduction in Section 7.4.2 as the main unverified link. read the letter →

arxiv 2501.18566 v2 pith:2UOKE4P2 submitted 2025-01-30 math.PR

Nicolas Curien , Grégory Miermont , Armand Riera This is my paper · ORCID
classification math.PR MSC 60D0560F1760G5205C8060G15
keywords BoltzmannplanarmapsstableLévyprocesseslooptreesscalinglimitsuniversalitySierpinskicarpetGaussianfreefieldHausdorffdimension
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 proves a universality theorem for random planar maps with very large faces. Fix α∈(1,2); any non-generic critical Boltzmann map whose root-face degree distribution has tail ~ c $k^{{-α}}$ converges, after graph distances are multiplied by (s_q n)^{-1/(2α)}, to a single random compact metric space S_α, the α-stable carpet for α≥3/2 and the α-stable gasket for α<3/2. The limit does not depend on the weight sequence q except through α, and has Hausdorff dimension 2α. This identifies a new one-parameter family of universal random geometries outside the Brownian sphere, and gives their topology: the Sierpinski carpet in the dilute phase, and a space whose faces may touch in the dense phase.

What carries the argument

The load-bearing objects are the α-stable looptree L coded by a spectrally positive α-stable Lévy excursion (loops glued along jumps), and the label process Z, defined as the Gaussian process on L with covariance given by the resistance metric — equivalently, Brownian motion indexed by the looptree. The proof works by (1) encoding discrete maps by labeled trees via a classical labeled-tree bijection, (2) passing to the coding-process limit (X,Z), (3) using fine properties of Z — absence of one-sided records on the skeleton, density of records on loops, and the exact two-point function — to prove that every subsequential limit identifies exactly the same points as the explicit pseudo-distance D*, and (4) a two-source labeled-tree construction to control geodesics between typical points and a surgery argument showing D and D* agree.

What would settle it

Take two non-generic weight sequences with the same α and simulate large conditioned Boltzmann maps; compute the rescaled Gromov–Hausdorff–Prokhorov limits (for example via distance matrices between a growing cloud of sampled vertices) and the Hausdorff dimension of the limiting space. If the two limits differ or the dimension is not 2α, Theorem 1.1 is false. A smaller-scale check: evaluate N(sup Z>1) by Monte Carlo on a discretized stable looptree with independent Brownian bridges on each loop; the paper's two-point function identities imply the exact value α(α−1)/2, so any discrepancy would trace the failure to the continuum construction.

Watch

Extended reading notes

Core claim

The central discovery is that the scaling limit of non-generic critical Boltzmann maps with exponent α exists and is described explicitly by a stable Lévy excursion X decorated by a Gaussian label process Z, the Brownian motion indexed by the stable looptree. The limiting space S_α is [0,1]/∼_{D*} with D* built from Z; the argument shows every subsequential limit D equals D* by identifying which points are glued — only trivial identifications in the looptree or zero-distance — and by showing geodesics between typical points are unique and can be compared via a two-source construction. En route the paper establishes fine quantitative facts: local minima of Z avoid the skeleton of the looptree; the two-point function satisfies N(sup Z>1)=α(α−1)/2; the volume of root-centered balls has stretched-exponential tails; geodesics to the root are simple, the cut locus is totally disconnected, and the maximal number of geodesics from a point is 2.

Load-bearing premise

The proof depends on the non-genericity condition (1.2), which requires the root-face degree tail to be asymptotic to a specific constant times $k^{{-α}}$ and to involve the same s_q appearing in the rescaling; if that fine-tuned condition fails, the limit is not claimed to be S_α.

Editorial extensions

If this is right

  • Every non-generic critical Boltzmann map with exponent α belongs to a single universality class: its scaling limit is S_α, independent of all other details of the face-weight sequence q.
  • The Hausdorff dimension 2α of S_α gives a new exact exponent for distances in these maps, refining the earlier tightness exponent.
  • The dichotomy at α=3/2 is sharp: for α∈[3/2,2) the limit is almost surely homeomorphic to the Sierpinski carpet, while for α∈(1,3/2) its faces touch, so the topology changes at the dilute/dense transition.
  • In any such limit, the geodesics toward a typical root point are simple, the cut locus is totally disconnected, and at most two distinct geodesics can start from the same point.
  • The volume of balls in S_α has stretched-exponential tail bounds, providing a robust local estimate usable in further metric-geometric analysis.

Reading between the lines

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

  • Not proved in the paper: the same convergence should hold if the non-genericity hypothesis is weakened to regular variation or if bipartiteness is dropped; the paper only sketches a re-rooting route toward these extensions.
  • Not proved in the paper: if the conjectural link to γ-LQG is correct, S_α should coincide with the chemical metric inside a conformal loop ensemble, giving these spaces a conformal-geometric characterization.
  • Not proved in the paper: as α→2, the construction should degenerate to the Brownian-sphere regime, interpolating between the stable and Brownian universality classes.
  • Not proved in the paper: the exact value N(sup Z>1)=α(α−1)/2 yields an explicit candidate for the two-point distance distribution, testable by simulation of the continuum process or of large maps.
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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 / 4 minor

Summary. The paper constructs, for each alpha in (1,2), a random compact metric measure space (S_alpha, D*_alpha, Vol_alpha) from a normalized alpha-stable Levy excursion decorated by independent Brownian bridges, and claims that large critical non-generic Boltzmann bipartite planar maps with weight sequence q of exponent alpha converge in distribution to this space after scaling distances by (s_q n)^{-1/(2 alpha)}. The claimed limit is universal, depending on q only through alpha, and has Hausdorff dimension 2 alpha. The introduction describes a proof strategy: tightness from [97]; identification of the point equivalence via faces; uniqueness of geodesics via a two-source construction; surgery along geodesics; and a priori ball-volume estimates. The paper also proves that in the dilute phase alpha in [3/2,2) the limit is homeomorphic to the Sierpinski carpet, while in the dense phase alpha in (1,3/2) faces may touch.

Significance. If Theorem 1.1 holds, this is a major advance: it establishes the first universal scaling limits for non-generic Boltzmann maps with large faces, constructs the stable carpets/gaskets, and develops a rich continuum theory (Gaussian free field on looptrees, exact two-point function, geodesic classification). The paper is notable for the exact computation N(sup Z > 1) = alpha(alpha-1)/2 via Bessel and hypergeometric functions, for the spinal decomposition of the label process, and for the topology theorems, including the Sierpinski carpet statement in the dilute phase. The introduction is exemplary in outlining a long and intricate proof. Credible strengths include the detailed treatment of the coding process and the robust a priori estimates leading to a dimension bound. The main caveat is that the proof of full Theorem 1.1 relies on deferred re-rooting and surgery arguments that are not verified in the material under review.

major comments (2)
  1. [Section 1 (after (1.5)); Section 7.4.2] Theorem 1.1 is stated for every admissible, critical, non-generic weight sequence q satisfying the tail condition (1.2). The text after (1.5) says that the proof is first carried out under the stronger pointwise condition w_q(deg(root face)=k) ~ C k^{-alpha-1}, and that the full statement follows from Le Gall's re-rooting trick, with details deferred to Section 7.4.2. Section 7.4.2 is not included in the material under review. This reduction is load-bearing: it must preserve the universal limiting space S_alpha and the normalization (s_q n)^{-1/(2 alpha)} for every q satisfying the fine-tuned condition (1.2). If the re-rooting argument requires additional regularity or yields a q-dependent limit or a different scaling constant, then Theorem 1.1 as stated is not established. Please provide the full details of Section 7.4.2, or restrict the statement of Theorem 1.1 to the pointwise condition and present the extension to the tail condition (1.2) as a conditional result.
  2. [Proof of Theorem 1.1; Sections 8-12] The introduction states that the equality D = D* for every subsequential limit is proven via the identification of equivalence classes (Theorem 8.1), the two-point construction with delays (Sections 10-11), the uniqueness of the typical geodesic (Theorem 9.1), and the surgery argument with bad-point estimates (Proposition 9.3 and Section 12). These sections are only summarized heuristically in the introduction and are not part of the reviewed text. Since these are the core steps that go beyond [97] and establish the uniqueness of the subsequential limit, the central claim of Theorem 1.1 cannot be fully verified from the supplied material. This is a structural verification gap rather than an identified mathematical error, and the authors should ensure that the complete proof is available in the version under review.
minor comments (4)
  1. [Index of notation] The entry for PMroot contains duplicated 'of' and a typo: 'isometry classes of of (rooted) iweighted geodesic compact metric spaces' should read 'isometry classes of (rooted) weighted geodesic compact metric spaces'.
  2. [Section 3.2, proof of Proposition 3.3] In several displays, the symbols 'eδ' and 'ed' are used where '~δ' and '~d' are intended (e.g., in the computation of the Brownian bridge covariance and in the final display of the proof). Please correct these typographical errors.
  3. [Section 5.2, proof of Proposition 5.4] The vanishing of the constants A' and B' in the hypergeometric expansion is asserted with 'a (tedious) computation shows that A'=B'=0', but the computation is not provided. Since the resulting value alpha(alpha-1)/2 determines the phase transition at alpha=3/2 and feeds into Proposition 5.6, please include the calculation in an appendix or give a precise reference.
  4. [Theorem 1.1 statement] The term 'admissible' in 'admissible, critical and non-generic weight sequence' is not defined in the introduction; please give a definition or a reference at the point of first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central contribution is the independent proof of uniqueness of subsequential limits, D = D*, built on prior external work rather than on its own conclusions.

full rationale

The derivation chain is incremental and non-circular. The maps are encoded by labeled mobiles via the Bouttier–Di Francesco–Guitter bijection; the scaling limit (X, Z) of these coding trees and the resulting tightness of the rescaled metric spaces are imported from Le Gall and Miermont [97]. That is a prior published result with stated assumptions that do not include Theorem 1.1, so it is legitimate external support even though one of the present authors is a co-author of [97]. The genuinely load-bearing new content is the proof that every subsequential limit metric D equals the explicitly constructed pseudo-distance D*: this is done through the point-identification theorem (Theorem 8.1), the two-point geodesic construction, uniqueness of typical geodesics, the strict upper bound on the dimension of bad points, and the surgery argument along geodesics. None of these steps is defined in terms of the target limit, and no fitted parameter is renamed as a prediction. The limit space S_alpha is built from the same stable excursion and Brownian bridges that appear in the coding-tree limit, but this is the standard structural route for Brownian-type scaling limits and does not make the conclusion equivalent to its inputs. The paper explicitly records that the non-genericity condition (1.2) is fine-tuned and that the reduction from the pointwise asymptotic to the tail condition via the re-rooting trick is deferred to Section 7.4.2; this is an omitted-proof/completeness concern about the full generality of Theorem 1.1, not a circularity. The computation of the constant alpha(alpha-1)/2 and the two-point function are self-contained continuum calculations. Overall, no circular step is identifiable in the reviewed material.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central result is a theorem with a proof, not a data-fitting exercise. No free parameters are fitted; alpha and s_q are model inputs. The proof relies on the non-genericity and bipartiteness assumptions on the map model, and on the prior scaling limit of labeled trees from [97]. No new physical or probabilistic entities are postulated beyond the explicitly constructed object S_alpha.

assumptions (3)
  • domain assumption The weight sequence q is non-generic with exponent alpha in (1,2), i.e. w_q(deg(root face) > k) ~ 2 s_q / |Gamma(1-alpha)| k^{-alpha} (eq. 1.2).
    This is the modeling assumption defining the class of maps under study. It is load-bearing for the scaling of the coding functions and for the entire proof; the theorem is only stated for maps satisfying it.
  • domain assumption All maps are bipartite, so faces have even degree and the BDG bijection applies.
    The proof uses the Bouttier-Di Francesco-Guitter bijection, which requires bipartiteness. Non-bipartite extensions are conjectured but not proven (see Remark 1.2).
  • standard math The coding functions of the labeled mobiles converge to (X,Z), the stable excursion and the label process, as established in Le Gall and Miermont [97].
    The paper takes as input the scaling limit of the labeled trees from [97]. This is a prior theorem, not proven in this paper, though the paper uses and extends it.

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Pith. "Pith review of The scaling limit of planar maps with large faces." pith.science (2026). https://pith.science/paper/2UOKE4P2

@misc{pith2026250118566,
  author       = {Pith},
  title        = {Pith review of: The scaling limit of planar maps with large faces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2UOKE4P2}},
  note         = {Machine review of arXiv:2501.18566}
}
abstract

We prove that large Boltzmann stable planar maps of index $\alpha \in (1;2)$ converge in the scaling limit towards a random compact metric space $\mathcal{S}_{\alpha}$ that we construct explicitly. They form a one-parameter family of random continuous spaces ``with holes'' or ``faces'' different from the Brownian sphere. In the so-called dilute phase $\alpha \in [3/2;2)$, the topology of $\mathcal{S}_{\alpha}$ is that of the Sierpinski carpet, while in the dense phase $\alpha \in (1;3/2)$ the ``faces'' of $\mathcal{S}_{\alpha}$ may touch each-others. En route, we prove various geometric properties of these objects concerning their faces or the behavior of geodesics.

Figures

Figures reproduced from arXiv: 2501.18566 by the authors.

Figure 1
Figure 1. Simulations of large non-generic critical random Boltzmann planar maps of index α ∈ {1.9, 1.8, 1.7, 1.6, 1.5, 1.4, 1.3} from top left to bottom right. Non-generic Boltzmann maps. Let us first introduce the model we will study in this paper, starting with some basic definitions. A planar map is a proper embedding of a finite multigraph in the two-dimensional sphere, such that the connected components of the complemen… view at source ↗
Figure 2
Figure 2. From top left to bottom right: The stable excursion X, the looptree L coded by X with colors indicating the loops, the label process Z, and finally the same looptree with colors indicating the values of the process Z. 0 ⩽ s < t ⩽ 1, set [t,s] := [0,s] ∪ [t, 1] and, for any s, t ∈ [0, 1], define z(s, t) := Zs + Zt − 2 max  min [s,t] Z, min [t,s] Z  , (1.7) and D ∗ (s, t) := inf p ∑ k=1 z(sk , tk) , (1.8) where the … view at source ↗
Figure 3
Figure 3. Illustration of the geometric underlying idea for the proof of D = 0 ⇐⇒ D∗ = 0. The “faces” of S are represented by the blue “holes”. Except for the trivial identifications, the distance D cannot identify more points since they must be separated by two faces. The proof of the latter result combines Moore’s theorem for quotients of the 2-dimensional sphere and a theorem of Whyburn which establishes that the Sierpinsk… view at source ↗
Figures from the paper (44 more)
Figure 4
Figure 4. Figure 4: Illustration of the neighborhood of a good point x ∈ γ1,2. The geodesic γ1,2 is drawn in red and blue. The faces F1 and F2 are drawn in light blue and green. The points ρ1 and ρ2 could be in the same connected component as x – even if we will see that this is not the s…
Figure 5
Figure 5. Figure 5: A simulation of a 3 2 -stable L´evy excursion and the corresponding looptree. Points belonging to the same loop (corresponding to the jumps of the excursion) are displayed with the same color. To lighten notation, for t ∈ (0, 1), we set At := {s ∈ [0, 1] : 0 ≼ s ≼ t an…
Figure 6
Figure 6. Figure 6: Illustration of times s < t such that s ∼d t. There are two possible cases: either X has a jump at time s, as depicted on the left, or X is continuous at s, and s is a local minimal record on the right, as depicted on the right. There are countably many pairs of identi…
Figure 7
Figure 7. Figure 7: An example of a loop (in orange), of pinch points (in red) and of two leaves (purple crosses) on a looptree. Next, we notice that (s, t) and (t,s) are connected, after identifying the points 0 and 1. Therefore, the continuity of Πd implies that C1 and C2 are two connec…
Figure 8
Figure 8. Figure 8: For a given t ∈ (0, 1), the times s ≼ t are the minimal records found by starting from Xt and following the running infimum of X backward in time (represented in red in the figure). These constitute the set Branch(0, t). Except at 0 and possibly at time t, all such tim…
Figure 9
Figure 9. Figure 9: Simulation of an α-stable tree and the associated looptree. The looptree L and the stable tree Th have “the same branching structure” except that the loops in L correspond to the branching points of infinite degree in Th. More precisely, by the definition of H and Prop…
Figure 10
Figure 10. Figure 10: Illustration of a portion of the processes X and H, both represented in the vicinity of a time t. A jump time for X (in orange) corresponds to a“plateau”of H on which it bounces. In particular, if r ≺ t is an ancestor of t at which ∆r > 0, we do not have ξt(Hr) = r. 3…
Figure 11
Figure 11. Figure 11: Simulation of the label process Z ∗ over [0, 1] (on the left) and seen on the looptree of [PITH_FULL_IMAGE:figures/full_fig_p037_11.png]
Figure 12
Figure 12. Figure 12: Illustration of the Markov property under Q. Conditionally on FT, the excursions of X above the running infimum starting at T (in red above), together with the shifted Z processes, form a Poisson process of intensity given in Lemma 3.5. Corollary 3.6. Let T be an (Ft)…
Figure 13
Figure 13. Figure 13: Illustration of the Markov property in terms of the looptree. We explore (in blue on the figure) in clockwise order a portion of the looptree –together with its labeling Z– until a stopping time T. The remaining pieces (in gray) attached to the “trunk”made of the (clo…
Figure 14
Figure 14. Figure 14: Illustration of the Markov property applied at time S n i : among the gray looptrees grafted on the 2 −n unit of length on the right of the trunk at time S n i (recall the caption of [PITH_FULL_IMAGE:figures/full_fig_p046_14.png]
Figure 15
Figure 15. Figure 15: The two times s and t are separated by the loop Πd ◦ fr([0, 1]). In blue are represented the set Πd([p1, p2]) and Πd([q1, q2]). the closest point of the loop from Πd [PITH_FULL_IMAGE:figures/full_fig_p048_15.png]
Figure 16
Figure 16. Figure 16: Illustration of the proof. At time Si the Ze-value is less than r. By scaling, in the forthcoming r 2α units of time, the variation of the infimum process I of X is of order r 2 , while one expects the variation of process Ze to be of order r. One way to ensure this i…
Figure 17
Figure 17. Figure 17: Illustration of the result of Proposition 4.5: Consider a branch in L and stop it at a pinch point Πd(t1) when reaching a new minimum for the label along that branch. Then we can find two blue loops on both sides “blocking” the level Zt1 . Let us describe first inform…
Figure 18
Figure 18. Figure 18: Illustration of R (i) t1,...,tm  1⩽i⩽m in the case m = 4. We are going to deduce Proposition 4.5 from the following result, which formalizes the above discussion: 55 [PITH_FULL_IMAGE:figures/full_fig_p055_18.png]
Figure 19
Figure 19. Figure 19: Illustration of the proof of Proposition 4.5. Left: finding a time t ′ 1 corresponds to finding a loop (in orange on the figure) blocking level z = Zt1 i.e. such that the label process along this loop takes values that are greater than z, and values that are less than…
Figure 20
Figure 20. Figure 20: A simulation of a Brownian motion (in blue) and its running infimum process (in orange). The trace is decorated with red slits which happen at time ti and extend over [inf Z i + b0(ti), b0(ti) + sup Z i ]. We are interested in the set of values B (in green on the left…
Figure 21
Figure 21. Figure 21: Illustration of an ε-trapped point X (in dark red on the figure). The spine between 0 and t• is made of the black loops. When following the looptree from the point X in one of the four directions, one encounters a (one-sided) minimal record time of Z corresponding to …
Figure 22
Figure 22. Figure 22: Illustration of Proposition 6.2: construction of the spine (and the dangling loop￾trees) from the process Y • . Proposition 6.2 (Spinal decomposition). Under N• and conditionally on r 7→ Y • r , the collection of random variables:  Sˇ r , Sr , Bˇ (r) , B (r) ,Pˇ r ,P…
Figure 23
Figure 23. Figure 23: Illustration of the definition of the random variables Rr and Ir . Here, only one side of the loops is displayed for visibility. The label on one side of the spine is obtained by concatenating the shifted process B (r) (in different colors above). The red slits corres…
Figure 21
Figure 21. Figure 21: Illustration of an #-trapped point X (in dark red on the figure). The spine between 0 and t • is made of the black loops. When following the looptree from the point X in one of the four directions, one encounters a (one-sided) minimal record time of Z corresponding to…
Figure 25
Figure 25. Figure 25: Illustration of the proof of C > 0 in Lemma 6.5: with a positive probability, the processes Y ↑ (in pink) and R (in green) may barely move over [0, ζ(r2)) and the first large jump at time θ produces a (1/r1)-trapping time. where the second inequality comes from Lemma …
Figure 26
Figure 26. Figure 26: Illustration of the Bouttier–Di Francesco–Guitter construction of a planar pointed map (in red on the right) from a well-labeled mobile (on the left). The pointed vertex is v ∗ and the orientation of the root edge is given by an independent sign ϵ. 99 [PITH_FULL_IMAG…
Figure 27
Figure 27. Figure 27: An example of a well-labeled mobile T = (T , ℓ) with the associated Lukasiewicz and label paths (S T , L T ). Heuristically, the white Lukasiewicz path encapsulates the tree structure of V◦(T ), and the label process represents the label function ℓ. These processes ar…
Figure 28
Figure 28. Figure 28: Illustration of the Cactus bound in the cactus representation of ([0, 1]/ ∼D, D). The vertical distances represent distances to ΠD(t∗). In red, we can see the two simple geodesics starting from ΠD(r1) and ΠD(r2) respectively. The bound follows by arguing that a D-geod…
Figure 29
Figure 29. Figure 29: Illustration of the cactus bound in the construction of maps from well-labeled mobiles. We draw in red the geodesics going from vr n 1 and vr n 2 to v∗ obtained in the construction of BDG bijection. In orange we draw the set Branch(vr n 1 , vr n 2 ). In the second cas…
Figure 30
Figure 30. Figure 30: Setup of the proof of Theorem 8.1 after excluding the trivial identifications d(s, t) = 0 and z(s, t) = 0. We can always find r1 ∈ (s, t) and r2 ∈ (s, t) pinch point times such that Zr1 < z, Zr2 < z and such that the labels on Branch(r1,r2)\{r1,r2} are strictly larger…
Figure 31
Figure 31. Figure 31: In the case when there exists r ∈ Branch(s, t)\{s, t} such that Zr = z, then such a time is unique and we can find r1 ∈ (t,s) and r2 ∈ (s, t) directly in the vicinity of r for the looptree distance d by Proposition 4.5. 8.2 Lamination encoding of the equivalence class…
Figure 32
Figure 32. Figure 32: Simulations of the geodesic laminations L(X) induced by ∼d (on the left) and L(Z) induced by ∼z (on the right). For clearness we have changed the Euclidean arcs [a, b]D by hyperbolic geodesics. Notice that the connected components of D\L(Z) are triangles and convex po…
Figure 33
Figure 33. Figure 33: An illustration of the images of L(X) in blue on the top hemisphere, and L(Z) in red, on the bottom hemisphere. which gives that S2/ ≈ is homeomorphic to S2. It remains to show that S1/ ≈ is homeomorphic to (S, D∗ ). In this direction, remark that by Theorem 8.1, for …
Figure 34
Figure 34. Figure 34: An illustration of the heuristic argument for the fact that the topology of S is sample-dependent in the dense case α ∈ (1, 3 2 ). Two touching faces (light and dark green) enclose a region separated by the two red extreme points. In the vicinity of each of these red …
Figure 35
Figure 35. Figure 35: Illustration of the covering of a face. The blacks dots correspond to the projection of the points f(s (ε) i ) and the white boxes to the projections of the points f(r (ε) i ). We can start simple geodesics (in red) from the two pre-images of Πd(f(s (ε) i )) and from …
Figure 36
Figure 36. Figure 36: Illustration of the proof: If ε is much smaller than ε ∗ , then we can find roughly N = ε ∗/ε ′ >> 1 points along a D-geodesic between x and y whose D∗ -balls of radius ε/3 are disjoint. The contradiction comes from the fact that BD(x, 2ε) (in blue on the figure) cont…
Figure 37
Figure 37. Figure 37: Top: illustration of ε-good points x. The geodesic γ1,2 is in red, while the geodesics towards ρ∗ are in green or yellow. Bottom: approximation of γ1,2 by pieces of geodesics towards ρ∗ using ε-good points. An a priori bound D∗ ⩽ D1−δ (Proposition 8.8) is used in the …
Figure 38
Figure 38. Figure 38: A unicyclomobile. is that the resulting map is a cycle, which must have even length by the bipartite nature of the map we started from. Conversely, it is a consequence of the Jordan curve theorem that a planar map with exactly one cycle has exactly two faces The sets …
Figure 39
Figure 39. Figure 39: The bi-pointed BDG construction applied on the unicyclomobile of [PITH_FULL_IMAGE:figures/full_fig_p133_39.png]
Figure 40
Figure 40. Figure 40: Decomposition of a mobile buckle into a mobile star and sub-mobiles, with k = 4 and r = 3 in the notation of the text. ℓ(vˆ◦) + ℓ ′ , and then identifying the leaf vˆ◦ of T with the root vertex of T ′ , in such a way that the corner incident to vˆ◦ is merged with the …
Figure 41
Figure 41. Figure 41: An element u ∈ U is described by a pair (T , vˆ◦),(P, vˆ ′ ◦ , c), where the first element (the belt) is a mobile with a marked leaf, and the second (the buckle) is a mobile buckle with a marked white corner c. The grey blobs indicate sub-mobiles, and circled grey blo…
Figure 42
Figure 42. Figure 42: Illustration of the notation. The (essentially unique) vertex minimizing the label along the cycle of umk is denoted by Jmk . The vertex xmk must be close to a vertex J ′ mk on the cycle which asymptotically minimizes the label. that d gr Mcmk (Jmk , J ′ mk ) = o(m 1 …
Figure 43
Figure 43. Figure 43: Illustration of the paths γ (um→vm) and γ (u ′ m→v ′ m) disconnecting respectively ΠD((um, vm)) and ΠD((u ′ m, v ′ m)) from its complement. A geodesic starting from ΠD(t) and targeting ρ∗ must either cross those paths or pass through ΠD(u ′ m) and ΠD(um), or ΠD(v ′ m)…
Figure 44
Figure 44. Figure 44: Illustration of the notation for Lemma 12.4. On this picture, y is an ε-bad point, because there exists a geodesic starting from the point z ′ ∈ Cε that immediately leaves γ1,2. Proof. Notice first that since there is a unique geodesic between x1 and x2, if a geodesic…
Figure 45
Figure 45. Figure 45: Illustration of the proof of Lemma 12.4. If any point of Cε is aligned with either {y (ε) 1 , x} or {y (ε) 2 , x}, then we can cover Cε with the range of two geodesics γ, γ ′ towards the root x. To conclude, it remains to show that if for every z ∈ Cε the points {z, y…
Figure 46
Figure 46. Figure 46: On the left, the ε-trapped condition for the point Xb imported from Section 6 and its geometric consequence on the right figure in (Sb, Db). The color code is the same in the left and right pictures, in particular, the geodesic γb1,2 is constructed by concatenating th…
Figure 47
Figure 47. Figure 47: Illustration of the proof of Lemma 12.8. On the event where Xb is ε-trapped, we can build in the discrete setting the analog of the blocking doors appearing in [PITH_FULL_IMAGE:figures/full_fig_p176_47.png]

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136 extracted references · 73 canonical work pages

  1. [97]

    Le Gall and G

    J.-F. Le Gall and G. Miermont. Scaling limits of random planar maps with large faces. Ann. Probab., 39(1):1–69, 2011

    work page 2011

  2. [1]

    Abraham, J.-F

    R. Abraham, J.-F. Delmas, and P. Hoscheit. A note on the Gromov-Hausdorff-Prokhorov dis- tance between (locally) compact metric measure spaces. Electron. J. Probab., 18, 2013

    2013

  3. [2]

    Conditioning (sub)critical L{\'e}vy trees by their maximal degree: Decomposition and local limit

    R. Abraham, J.-F. Delmas, and M. Nassif. Conditioning (sub)critical L´ evy trees by their maximal degree: Decomposition and local limit. arXiv:2211.02317, 2022

    work page Pith review arXiv 2022

  4. [3]

    Addario-Berry and M

    L. Addario-Berry and M. Albenque. The scaling limit of random simple triangulations and random simple quadrangulations. Ann. Probab., 45(5):2767–2825, 2017

    2017

  5. [4]

    Geometric properties of spin clusters in random triangulations coupled with an Ising Model

    M. Albenque and L. M´ enard. Geometric properties of spin clusters in random triangulations coupled with an Ising model. arXiv:2201.11922, 2022. 178

    work page Pith review arXiv 2022

  6. [5]

    Ambjørn, T

    J. Ambjørn, T. Budd, and Y. Makeenko. Generalized multicritical one-matrix models. Nuclear Physics B , 913:357–380, 2016

    2016

  7. [6]

    Ambrosio, J

    V. Ambrosio, J. Miller, and Y. Yuan. Chemical distance metric for non-simple CLE. (in preparation), 2024

    2024

  8. [7]

    G. E. Andrews, R. Askey, and R. Roy. Special Functions, volume 71 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999

    1999

Show all 136 references
  1. [8]

    Angel, B

    O. Angel, B. Kolesnik, and G. Miermont. Stability of geodesics in the Brownian map. Ann. Probab., 45(5):3451–3479, 2017

    2017

  2. [9]

    E. Archer. Brownian motion on stable looptrees. In Annales de l’Institut Henri Poincar´ e- Probabilit´ es et Statistiques, volume 57, pages 940–979, 2021

    2021

  3. [10]

    Archer, A

    E. Archer, A. Carrance, and L. M´ enard. Some properties of stable snakes. arXiv:2403.15275, 2024

    2024 arXiv

  4. [11]

    Archer, A

    E. Archer, A. Carrance, and L. M´ enard. Stable quadrangulations and stable spheres. arXiv:2405.05677, 2024

    2024 arXiv

  5. [12]

    E. W. Barnes. The genesis of the double Gamma function. Proceedings of the London Mathe- matical Society, 31:358–381, 1899

  6. [13]

    E. W. Barnes. The theory of the double Gamma function. Philosophical Transactions of the Royal Society of London (A) , 196:265–387, 1901

    1901

  7. [14]

    Berestycki

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

    2015

  8. [15]

    Bernardi, N

    O. Bernardi, N. Curien, and G. Miermont. A Boltzmann approach to percolation on random triangulations. Canadian Journal of Mathematics , 71(1):1–43, 2019

    2019

  9. [16]

    J. Bertoin. Increase of stable processes. J. Theoret. Probab., 7(3):551–563, 1994

    1994

  10. [17]

    J. Bertoin. L´ evy processes, volume 121 of Cambridge Tracts in Mathematics . Cambridge Uni- versity Press, Cambridge, 1996

    1996

  11. [18]

    J. Bertoin. Subordinators: examples and applications. In Lectures on probability theory and statistics (Saint-Flour, 1997) , volume 1717 of Lecture Notes in Math. , pages 1–91. Springer, Berlin, 1999

    1997

  12. [19]

    Bertoin, T

    J. Bertoin, T. Budd, N. Curien, and I. Kortchemski. Martingales in self-similar growth- fragmentations and their connections with random planar maps. Probab. Theory Related Fields, 172(3):663–724, 2018. 179

    2018

  13. [20]

    Bertoin, N

    J. Bertoin, N. Curien, and A. Riera. Self-similar markov trees and scaling limits. arXiv:2407.07888, 2024

    2024 arXiv

  14. [21]

    Bertoin and M

    J. Bertoin and M. Savov. Some applications of duality for L´ evy processes in a half-line. Bulletin of the London Mathematical Society , 43(1):97–110, 10 2010

    2010

  15. [22]

    Bettinelli

    J. Bettinelli. Scaling limit of random planar quadrangulations with a boundary. In Annales de l’IHP Probabilit´ es et statistiques, volume 51, pages 432–477, 2015

    2015

  16. [23]

    Bettinelli

    J. Bettinelli. Geodesics in Brownian surfaces (Brownian maps). Ann. Inst. Henri Poincar´ e Probab. Stat., 52(2):612–646, 2016

    2016

  17. [24]

    Bettinelli, E

    J. Bettinelli, E. Jacob, and G. Miermont. The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection. Electronic J. Probab., 19, 2014

    2014

  18. [25]

    Bettinelli and G

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

    2017

  19. [26]

    Bettinelli and G

    J. Bettinelli and G. Miermont. Compact Brownian surfaces II. Orientable surfaces. arXiv:2212.12511, 2022

    2022

  20. [27]

    Bj ¨ornberg, N

    J. Bj ¨ornberg, N. Curien, and S. ¨O. Stef´ ansson. Stable shredded spheres and causal random maps with large faces. Ann. of Probab., 50(5):2056–2084, 2022

    2022

  21. [28]

    Blanc, N

    G. Blanc, N. Curien, and J. Kahn. Geodesics in planar Poisson roads random metric. arXiv:2407.07887, 2024

    2024 arXiv

  22. [29]

    Blanc-Renaudie

    A. Blanc-Renaudie. Looptree, Fennec, and Snake of ICRT. arXiv:2203.10891, 2022

    2022 arXiv

  23. [30]

    Borot, J

    G. Borot, J. Bouttier, and E. Guitter. Loop models on random maps via nested loops: case of domain symmetry breaking and application to the potts model. J. Phys. A: Math. Theor. , 45(49), 2012

    2012

  24. [31]

    Borot, J

    G. Borot, J. Bouttier, and E. Guitter. A recursive approach to the O(N) model on random maps via nested loops. J. Phys. A: Math. Theor. , 45, 2012

    2012

  25. [32]

    Bouttier, P

    J. Bouttier, P. Di Francesco, and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin., 11(1):Research Paper 69, 27 pp. (electronic), 2004

    2004

  26. [33]

    Bouttier, E

    J. Bouttier, E. Guitter, and G. Miermont. Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants. Annales Henri Lebesgue , 5:1035–1110, 2022

    2022

  27. [34]

    Bretagnolle

    J. Bretagnolle. Sur l’in´ egalit´ e de concentration de Doeblin-L´ evy, Rogozin-Kesten. InParametric and semiparametric models with applications to reliability, survival analysis, and quality of life , Stat. Ind. Technol., pages 533–551. Birkh ¨auser Boston, Boston, MA, 2004. 180

    2004

  28. [35]

    Budd and L

    T. Budd and L. Chen. The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen). Electron. J. Probab., 2019

    2019

  29. [36]

    Budd and N

    T. Budd and N. Curien. Geometry of infinite planar maps with high degrees. Electron. J. Probab., 22:Paper No. 35, 37, 2017

    2017

  30. [37]

    Budd and N

    T. Budd and N. Curien. Random punctured hyperbolic surfaces & the Brownian sphere. (in preparation), 2024

    2024

  31. [38]

    T. Budd, N. Curien, and C. Marzouk. Infinite random planar maps related to Cauchy processes. J. ´Ec. polytech. Math., 5:749–791, 2018

    2018

  32. [39]

    Burago, Y

    D. Burago, Y. Burago, and S. Ivanov. A course in metric geometry , volume 33 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2001

    2001

  33. [40]

    Caraceni and N

    A. Caraceni and N. Curien. Self-avoiding walks on the UIPQ. In Sojourns in Probability Theory and Statistical Physics-III , pages 138–165. Springer, 2019

    2019

  34. [41]

    Chassaing and G

    P. Chassaing and G. Schaeffer. Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields, 128(2):161–212, 2004

    2004

  35. [42]

    Chaumont

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

    1996

  36. [43]

    Chaumont and G

    L. Chaumont and G. Uribe Bravo. Shifting processes with cyclically exchangeable increments at random. XI Symposium on Probability and Stochastic Processes , (69):101–117, 2015

    2015

  37. [44]

    Chen and J

    L. Chen and J. Turunen. Critical ising model on random triangulations of the disk: enumeration and local limits. Communications in Mathematical Physics , 374(3):1577–1643, 2020

    2020

  38. [45]

    Chen and J

    L. Chen and J. Turunen. Ising model on random triangulations of the disk: phase transition. Communications in Mathematical Physics , 397(2):793–873, 2023

    2023

  39. [46]

    N. Curien. Peeling Random Planar Maps: ´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour XLIX– 2019, volume 2335. Springer Nature, 2023

    2019

  40. [47]

    Curien and I

    N. Curien and I. Kortchemski. Random stable looptrees. Electron. J. Probab., 19:no. 108, 35, 2014

    2014

  41. [48]

    Curien and J.-F

    N. Curien and J.-F. Le Gall. Random recursive triangulations of the disk via fragmentation theory. Ann. Probab., 39(6):2224–2270, 2011

    2011

  42. [49]

    Curien and J.-F

    N. Curien and J.-F. Le Gall. The Brownian plane. J. Theoret. Probab., 27(4):1249–1291, 2014

    2014

  43. [50]

    Curien and J.-F

    N. Curien and J.-F. Le Gall. First-passage percolation and local perturbations on random planar maps. Ann. Sci. ´Ec. Norm. Sup´ er, 3(52):631–701, 2019. 181

    2019

  44. [51]

    Curien, J.-F

    N. Curien, J.-F. Le Gall, and G. Miermont. The Brownian cactus I. Scaling limits of discrete cactuses. Ann. Inst. Henri Poincar´ e Probab. Stat., 49(2):340–373, 2013

    2013

  45. [52]

    Curien, L

    N. Curien, L. M´ enard, and G. Miermont. A view from infinity of the uniform infinite planar quadrangulation. ALEA Lat. Am. J. Probab. Math. Stat. , 10(1):45–88, 2013

    2013

  46. [53]

    Curien and L

    N. Curien and L. Richier. Duality of random planar maps via percolation. InAnnales de l’Institut Fourier, volume 70, pages 2425–2471, 2020

    2020

  47. [54]

    Dauvergne

    D. Dauvergne. The 27 geodesic networks in the directed landscape. arXiv:2302.07802, 2023

    2023 arXiv

  48. [55]

    J.-F. Delmas. Computation of moments for the length of the onedimensional ISE support. Electron. J. Probab., 8:1–15, 2003

    2003

  49. [56]

    Doherty, K

    C. Doherty, K. Kavvadias, and J. Miller. Connectivity of the adjacency graph of complementary components of the SLE fan. arXiv:2411.13133, 2024

    2024 arXiv

  50. [57]

    Duplantier, J

    B. Duplantier, J. R. Miller, and S. Sheffield. Liouville quantum gravity as a mating of trees. Asterisque, 427, 2021

    2021

  51. [58]

    Duquesne

    T. Duquesne. A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab., 31(2):996–1027, 2003

    2003

  52. [59]

    Duquesne and J.-F

    T. Duquesne and J.-F. Le Gall. Random trees, L´ evy processes and spatial branching processes. Ast´ erisque, (281):vi+147, 2002

    2002

  53. [60]

    Duquesne and J.-F

    T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of L´ evy trees. Probab. Theory Related Fields, 131(4):553–603, 2005

    2005

  54. [61]

    P. J. Fitzsimmons, B. Fristedt, and L. A. Shepp. The set of real numbers left uncovered by random covering intervals. Z. Wahrsch. Verw. Gebiete , 70(2):175–189, 1985

    1985

  55. [62]

    Garban, R

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

    2016

  56. [63]

    R. K. Getoor. Excursions of a Markov Process. Ann. of Probab., 7(2):244 – 266, 1979

    1979

  57. [64]

    Topological characterization of the Sierpinski curve

    G.Whyburn. Topological characterization of the Sierpinski curve. Fund. Math. 45 , 1958

    1958

  58. [65]

    E. Gwynne. Geodesic networks in Liouville quantum gravity surfaces. Probab. Math. Phys. , 2(3):643–684, 2021

    2021

  59. [66]

    Gwynne, N

    E. Gwynne, N. Holden, and X. Sun. Mating of trees for random planar maps and Liouville quantum gravity: a survey. arXiv:1910.04713, 2019

    1910 arXiv

  60. [67]

    Gwynne and J

    E. Gwynne and J. Miller. Convergence of percolation on uniform quadrangulations with bound- ary to SLE 6 on √ 8/3-Liouville quantum gravity. arXiv:1701.05175, 2017. 182

    2017 arXiv

  61. [68]

    Gwynne and J

    E. Gwynne and J. Miller. Existence and uniqueness of the Liouville quantum gravity metric for γ ∈ (0, 2). Invent. Math., pages 1–121, 2020

    2020

  62. [69]

    Gwynne and J

    E. Gwynne and J. Miller. Convergence of the self-avoiding walk on random quadrangulations to SLE8/3 on √ 8/3-Liouville quantum gravity. Ann. Sci. ´Ec. Norm. Sup´ er., 54:305–405, 2021

    2021

  63. [70]

    Gwynne, J

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

    2021

  64. [71]

    Gwynne and J

    E. Gwynne and J. Pfeffer. Connectivity properties of the adjacency graph of SLE κ bubbles for κ ∈ (4, 8). Ann. of Probab., 48(3):1495–1519, 2020

    2020

  65. [72]

    J. Hawkes. Intersections of markov random sets. Zeitschrift f ¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 37(3):243–251, 1977

    1977

  66. [73]

    I. A. Ibragimov and Y. V. Linnik. Independent and stationary sequences of random variables . Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibrag- imov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman

    1971

  67. [74]

    Kammerer

    E. Kammerer. Scaling limit of first passage percolation geodesics on planar maps. arXiv:2412.02666

    arXiv

  68. [75]

    Kammerer

    E. Kammerer. Distances on the CLE4, critical Liouville quantum gravity and 3/2-stable maps. arXiv:2311.08571, 2024

    2024

  69. [76]

    Kammerer

    E. Kammerer. Gaskets of O(2) loop-decorated random planar maps. arXiv:2411.05541, 2024

    2024

  70. [77]

    Kammerer

    E. Kammerer. On large 3/2-stable maps. Annales de l’Institut Henri Poincar´ e, To appear

  71. [78]

    R. Khanfir. Convergences of looptrees coded by excursions. Annales de l’Institut Henri Poincar´ e, To appear

  72. [79]

    A. Khezeli. Metrization of the Gromov–Hausdorff (-Prokhorov) topology for boundedly-compact metric spaces. Stochastic Processes and their Applications , 130(6):101–117, 2020

    2020

  73. [80]

    A. Khezeli. A unified framework for generalizing the Gromov-Hausdorff metric. Probability Surveys, 20:837–896, 2023

    2023

  74. [81]

    Kortchemski

    I. Kortchemski. Random stable laminations of the disk. Ann. of Probab., 42(2):725–759, 2014

    2014

  75. [82]

    Kortchemski and C

    I. Kortchemski and C. Marzouk. Random L´ evy Looptrees and L´ evy maps. arXiv:2402.04098, 2024

    2024 arXiv

  76. [83]

    Kuratowski

    K. Kuratowski. Topology II. Academic Press, New York, (1968)

    1968

  77. [84]

    Kuznetsov

    A. Kuznetsov. On extrema of stable processes. Ann. of Probab., 39:1027–1060, 2011. 183

    2011

  78. [85]

    A. E. Kyprianou and J. C. Pardo. Stable L´ evy processes via Lamperti-type representations, volume 7. Cambridge University Press, 2022

    2022

  79. [86]

    S. K. Lando and A. Zvonkin. Graphs on surfaces and their applications . Springer-Verlag, 2004

    2004

  80. [88]

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

    1999

  81. [89]

    J.-F. Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math. , 169(3):621–670, 2007

    2007

  82. [90]

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

    2010

  83. [91]

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

    2013

  84. [92]

    J.-F. Le Gall. Geodesics stars in random geometry. Ann. of Probab., 50:1013–1058, 2022

    2022

  85. [93]

    J.-F. Le Gall. The volume measure of the Brownian sphere is a Hausdorff measure. Electron. J. Probab., 27:1–28, 2022

    2022

  86. [94]

    Le Gall and Y

    J.-F. Le Gall and Y. Le Jan. Branching processes in L´ evy processes: the exploration process. Ann. Probab., 26(1):213–252, 1998

    1998

  87. [95]

    Le Gall and T

    J.-F. Le Gall and T. Leh´ ericy. Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation. Ann. Probab., 47:1498–1540, 2019

    2019

  88. [96]

    Le Gall and A

    J.-F. Le Gall and A. Metz-Donnadieu. Drilling holes in the Brownian disk: The Brownian annulus. arXiv:2407.13544, 2024

    2024 arXiv

  89. [98]

    Le Gall and F

    J.-F. Le Gall and F. Paulin. Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal., 18(3):893–918, 2008

    2008

  90. [99]

    Le Gall and A

    J.-F. Le Gall and A. Riera. Some explicit distributions for Brownian motion indexed by the Brownian tree. Markov Processes Relat. Fields, 26:659–686, 2020

    2020

  91. [100]

    Le Gall and A

    J.-F. Le Gall and A. Riera. Spine representations for non-compact models of random geometry. Probab. Theory Related Fields, 181(1):571–645, 2021

    2021

  92. [101]

    Le Gall and A

    J.-F. Le Gall and A. Riera. Peeling the Brownian half-plane. arXiv:2404.18489, 2024. 184

    2024 arXiv

  93. [102]

    Le Gall and A

    J.-F. Le Gall and A. Riera. Spatial Markov property in Brownian disks. Annales de l’Institut Henri Poincar´ e, To appear

  94. [103]

    Lyons and Y

    R. Lyons and Y. Peres. Probability on Trees and Networks , volume 42 of Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge University Press, New York, 2016. Available at http://pages.iu.edu/~rdlyons/

    2016

  95. [104]

    Maisonneuve

    B. Maisonneuve. On the structure of certain excursions of a Markov process. Z. Wahrschein- lichkeitstheorie verw Gebiete , 47:61–67, 1979

    1979

  96. [105]

    Marckert and G

    J.-F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab., 35(5):1642–1705, 2007

    2007

  97. [106]

    Marckert and A

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

    2006

  98. [107]

    M. B. Marcus and J. Rosen. Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2006

    2006

  99. [108]

    C. Marzouk. On scaling limits of planar maps with stable face-degrees. ALEA, 15:1089–1122, 2018

    2018

  100. [109]

    C. Marzouk. On scaling limits of random trees and maps with a prescribed degree sequence. Annales Henri Lebesgue, 5:317–386, 2022

    2022

  101. [110]

    Miermont

    G. Miermont. On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab., 13:248–257, 2008

    2008

  102. [111]

    Miermont

    G. Miermont. Tessellations of random maps of arbitrary genus. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 42(5):725–781, 2009

    2009

  103. [112]

    Miermont

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

    2013

  104. [113]

    J. Miller. Tightness of approximations to the chemical distance metric for simple Conformal Loop Ensembles. arXiv:2112.08335, 2021

    2021 arXiv

  105. [114]

    Miller and W

    J. Miller and W. Qian. Geodesics in the Brownian map: Strong confluence and geometric structure. arXiv:2008.02242, 2020

    2008 arXiv

  106. [115]

    Miller and S

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

    2016

  107. [116]

    Miller and S

    J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map I: The QLE (8/3,

  108. [117]

    metric. Invent. Math., 219(1):75–152, 2020. 185

    2020

  109. [118]

    Miller and S

    J. Miller and S. Sheffield. An axiomatic characterization of the brownian map. Journal de l’ ´Ecole polytechnique—Math´ ematiques, 8:609–731, 2021

    2021

  110. [119]

    Miller and S

    J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding. Ann. of Probab., 49(6):2732–2829, 2021

    2021

  111. [120]

    Miller and S

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

    2021

  112. [121]

    Miller, S

    J. Miller, S. Sheffield, and W. Werner. CLE percolations. InForum of Mathematics, Pi, volume 5. Cambridge University Press, 2017

    2017

  113. [122]

    R. Moore. Concerning upper-semicontinuous collections of continua. Amer.Math.Soc, 1925

    1925

  114. [123]

    M ¨orters and Y

    P. M ¨orters and Y. Peres. Brownian motion . Cambridge Series in Statistical and Probabilis- tic Mathematics. Cambridge University Press, Cambridge, 2010. With an appendix by Oded Schramm and Wendelin Werner

    2010

  115. [124]

    Mourichoux

    M. Mourichoux. The bigeodesic Brownian plane. arXiv:2410.00426, 2024

    2024 arXiv

  116. [125]

    V. V. Petrov. Sums of independent random variables , volume Band 82 of Ergebnisse der Mathe- matik und ihrer Grenzgebiete [Results in Mathematics and Related Areas] . Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown

    1975

  117. [126]

    Revuz and M

    D. Revuz and M. Yor. Continuous martingales and Brownian motion , volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci- ences]. Springer-Verlag, Berlin, third edition, 1999

    1999

  118. [127]

    A. Riera. Isoperimetric inequalities in the Brownian map and the Brownian plane. Ann. Probab., 50:2013–2055, 2022

    2013

  119. [128]

    Riera and A

    A. Riera and A. Rosales-Ortiz. Excursion theory for Markov processes indexed by Levy trees. arXiv:2411.12717, 2024

    2024 arXiv

  120. [129]

    Rohde and O

    S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. (2) , 161(2):883–924, 2005

    2005

  121. [130]

    Schaeffer

    G. Schaeffer. Conjugaison d’arbres et cartes combinatoires al´ eatoires. PhD thesis. 1998

    1998

  122. [131]

    Sheffield

    S. Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. , 147(1):79–129, 2009

    2009

  123. [132]

    Sheffield

    S. Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab., pages 3474–3545, 2016

    2016

  124. [133]

    Sheffield

    S. Sheffield. Quantum gravity and inventory accumulation. Ann. Probab., 44(6):3804–3848, 2016. 186

    2016

  125. [134]

    Sheffield and W

    S. Sheffield and W. Werner. Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. of Math. , 176(3):1827–1917, 2012

    1917

  126. [135]

    V. Vigon. Votre L´ evy rampe-t-il?Journal of the London Mathematical Society , 65(1):243 – 256, 2002

    2002

  127. [136]

    Yearwood

    S. Yearwood. The topology of SLEκ is random for κ > 4. Electron. J. Probab. , 27:Paper No. 156, 14, 2022

    2022

  128. [137]

    V. M. Zolotarev. One-dimensional Stable Distributions , volume 65. American Mathematical Society, translations of mathematical monographs edition, 1986. 187

    1986

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