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Random infinite ideal angled graphs and ideal hyperbolic polyhedra

T0 review · 5 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The expected sum of dihedral angles around a vertex decides whether a random infinite ideal hyperbolic polyhedron is parabolic or hyperbolic — and in the hyperbolic case the random walk escapes to a well-defined boundary at positive speed.

desk verdict Serious extension of AHNR to ideal circle packings with a nice new invariant, but two proofs have load-bearing gaps that need fixing before this is citable. read the letter →

arxiv 2601.14909 v2 pith:RH4HWEGR submitted 2026-01-21 math.PR math.COmath.CVmath.DGmath.GT

Huabin Ge , Yangxiang Lu , Chuwen Wang , Tian Zhou This is my paper
classification math.PRmath.COmath.CVmath.DGmath.GT MSC 60K3752C2651M10
keywords idealangledgraphshyperbolicpolyhedracirclepackingsunimodularrandomdichotomytheoremPoissonboundarywalkringlemma
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 establishes a dichotomy for random infinite ideal polyhedra in hyperbolic 3-space, equivalently for random ideal circle packings via their ideal angled graphs. For an ergodic unimodular random IAG with finite expected degree, the expected angle character E[T(ρ)] = E[Σ_{e∋ρ} Θ_e] is always at least 2π; it equals 2π exactly when the structure is almost surely parabolic — invariantly amenable, recurrent, admitting a circle packing in the plane — and exceeds 2π exactly when it is almost surely hyperbolic — invariantly non-amenable, transient, admitting a packing in the unit disk. In the hyperbolic case the simple random walk converges almost surely to a boundary point of the unit disk, the exit measure is non-atomic with full support, the disk boundary realizes the Poisson boundary, and the walk escapes with positive linear hyperbolic speed equal to the exponential decay rate of circle radii. The result extends the known dichotomy for planar triangulations to the much larger class of angle-weighted planar graphs by proving a refined ring lemma for ideal circle packings.

What carries the argument

The central object is the angle character T(ρ) := Σ_{e∋ρ} Θ_e, the sum of dihedral angles of the ideal polyhedron at the face corresponding to the root vertex, which plays the role that combinatorial curvature plays for triangulations. The paper's main new estimate is a Refined Ring Lemma: for a tame IAG with angles in [0, π−ε1] and the strengthened Rivin condition (C′2), the radius ratio r(v)/r(u) across any edge u~v is at least e^{−C·S(u)}, where S(u)=Σ_{v′∼u} deg(v′) is the flower degree and C depends only on the angle bounds. This exponential local control of radii is what converts exponential decay of radii along the induced walk on an invariant percolation cluster into exponential deca

What would settle it

Compute, for a tame IAG with embedded ICP and angles bounded away from π, the ratio r(v)/r(u) along an edge where the flower degree S(u) is large; the Refined Ring Lemma predicts log(r(v)/r(u)) ≥ −C·S(u) with C depending only on the angle bounds. A concrete example with log(r(v)/r(u)) < −C·S(u) for a fixed C would falsify the lemma and, with it, the positive-speed and boundary-convergence theorems. Alternatively, an ergodic unimodular IAG with E[T(ρ)]=2π that is transient, or E[T(ρ)]>2π that is recurrent, would falsify the dichotomy.

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Extended reading notes

Core claim

The central claim is that for unimodular random ideal angled graphs the expected angle character E[T(ρ)] is a complete invariant determining the ideal-circle-packing type: E[T(ρ)] = 2π if and only if the graph is almost surely ICP-parabolic (equivalently invariantly amenable, VEL-parabolic, recurrent), and E[T(ρ)] > 2π if and only if it is almost surely ICP-hyperbolic (equivalently invariantly non-amenable, VEL-hyperbolic, transient). In the hyperbolic case, assuming a finite second moment of degree and angles uniformly bounded away from π, the Euclidean and hyperbolic circle centers z(X_n) and z_h(X_n) converge almost surely to a common boundary point δ on ∂D; the exit measure is non-atomic

Load-bearing premise

The load-bearing premise is the Refined Ring Lemma, whose claim is that in a tame ideal circle packing with angles uniformly bounded away from π, the radius of a neighboring circle can shrink by at most an exponential factor in the vertex's flower degree; the paper's proof of this lemma is the most compressed step, and if this radius-ratio bound fails, the exponential decay of radii, the boundary convergence, and the positive-speed theorem lose their engine.

Editorial extensions

If this is right

  • For any ergodic unimodular random IAG with finite expected degree, the single value E[T(ρ)] decides whether the associated random ideal polyhedron is parabolic (planar, recurrent, amenable) or hyperbolic (disk, transient, non-amenable).
  • In the hyperbolic case, the random walk converges almost surely to a point on the unit circle, so the circle boundary gives a concrete tail description of the walk.
  • The exit measure on the boundary has full support and no atoms, so every open arc is hit with positive probability and no single boundary point carries positive mass.
  • The walk's hyperbolic displacement grows linearly at a positive rate, and this rate is exactly the exponential decay rate of the radii of the visited circles.
  • The dichotomy and boundary coincidences extend to all tame ideal angled graphs, not just triangulations, so the same statements hold for the corresponding ideal hyperbolic polyhedra.

Reading between the lines

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

  • If the dichotomy is correct, the expected angle character E[T(ρ)] becomes a computable invariant for classifying random planar angle-weighted graphs as hyperbolic or parabolic, and could serve as a geometric order parameter for phase transitions in random polyhedral models.
  • The refined ring lemma suggests a general phenomenon for circle packings with prescribed intersection angles: local radius ratios are exponentially controlled by local degree structure; this may extend to non-ideal circle patterns or circle patterns on surfaces with cone singularities.
  • The authors conjecture that E[deg(ρ)] < ∞ (or a higher moment) suffices for Martin-boundary identification; if true, the bounded-degree hypothesis in the boundary theory could be relaxed substantially.
  • The equality between hyperbolic speed and logarithmic radius decay gives a new way to measure the asymptotic geometry of random ideal polyhedra from the circle packing alone, and might be testable numerically on concrete packings.
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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

5 major / 5 minor

Summary. The paper develops a boundary and dichotomy theory for random infinite ideal angled graphs (IAGs), i.e., weighted planar graphs (G,Θ) arising as dual 1-skeleta of ideal hyperbolic polyhedra, and equivalently for ideal circle packings (ICPs). The main claims are: (1) for an ergodic unimodular random tame IAG with finite expected degree, the expected angle character E[T(ρ)] is always at least 2π, with equality characterizing invariant amenability, VEL-parabolicity, ICP-parabolicity, and recurrence, and strict inequality characterizing the hyperbolic/transient side (Theorem 1.2 and Corollary 1.3); (2) in the ICP-hyperbolic case under E[deg²(ρ)]<∞ and an angle upper bound, the simple random walk converges a.s. to a point on ∂D, the exit measure is non-atomic with full support, ∂D realizes the Poisson boundary, and the hyperbolic speed is positive and equals the exponential decay rate of circle radii (Theorems 1.4 and 1.5). The proofs use a mass-transport identity for E[T]=E[θ], a geometric dichotomy based on the AHNR framework, a refined ring lemma for ICPs, and an induced-walk argument with exponential decay of radii.

Significance. If the main theorems are correct, the paper would substantially extend the Angel–Hutchcroft–Nachmias–Ray theory from triangulations and general planar maps to angle-weighted ideal circle packings and ideal hyperbolic polyhedra, providing a single geometric invariant — the expected angle character — that detects amenability, conformal type, recurrence/transience, and boundary identification. The conceptual framework is attractive, and the mass-transport proof of Theorem 1.1 is explicit and honest: E[T(ρ)]=E[θ(ρ)] is derived cleanly from a concrete unimodular transport. The paper also identifies the refined ring lemma as the key bridge from local geometry to global random-walk behavior, which is the right structural insight. However, as it stands the central proof contains several load-bearing gaps: the dichotomy proof is partly self-referential and leaves the angle identities unchecked; the counting lemma used for exponential decay is false as stated for arbitrary disc families; and the Birkhoff step in the exponential-decay and speed theorems requires a moment control that is not provided by E[deg²(ρ)]<∞. These issues touch Theorems 1.2, 1.4, and 1.5 directly, so the current manusc

major comments (5)
  1. [§3.2, Eqs. (10)–(13)] The proof of Theorem 1.2 opens with ‘by The Dichotomy Theorem 1.2’, which is a self-reference unless an external theorem from [4,5] is intended; this must be stated explicitly and its hypotheses checked. More importantly, Eqs. (10)–(13) assert that the mass transport F has outgoing total 4π and incoming total 2T(ρ) in the parabolic case and <2T(ρ) in the hyperbolic case. These identities are not derived. In particular, the notation f_i=(v_i,ρ,v_{i+1}) in (11) appears to assume that the faces incident to ρ are triangles, whereas IAGs are defined for arbitrary cellular decompositions. This is the central step establishing E[T(ρ)]=2π iff ICP-parabolic; it needs a complete angle bookkeeping for faces of arbitrary length, or a precise reduction to [5].
  2. [§4.2, Lemma 4.8] Lemma 4.8 is false as stated for arbitrary families of discs in D: taking countably many discs with centers in a small neighbourhood and radii 1/2 gives N(τ)=∞ while Cτ^{-2} is finite. The proof claims a decomposition into at most Q pairwise-disjoint subfamilies using Besicovitch covering; Besicovitch provides a bounded number of covering subfamilies, not a partition of the index set. The subsequent bound #{v: r(v)≥e^{-cm/4}} ≤ C e^{cm/2} and hence the Borel–Cantelli step in (21) depend on this lemma. A correct counting argument must use the specific geometric constraints of an embedded ICP, e.g. disjointness of the associated quadrilaterals or a separation property following from planarity, rather than an arbitrary disc-family estimate.
  3. [§4.2, after Eq. (27), and §5, Eq. (37)] The proof applies Birkhoff’s ergodic theorem to S(X_i) under the reversible/stationary law. It cites Lemma 4.9 for E[S(ρ)]<∞, but Lemma 4.9 computes the expectation under the original unimodular measure. Under the degree-biased reversible measure, E_rev[S(ρ)] = E[deg(ρ)S(ρ)] / E[deg(ρ)], and E[deg(ρ)S(ρ)] is not bounded by E[deg²(ρ)]; it is a cross-moment involving deg(ρ) times the sum of neighbour degrees. Thus the claimed a.s. vanishing in (29) is not justified. The same missing integrability appears in Theorem 5.1, where E_rev[|log R_1|] ≤ A E_rev[S(ρ)] is asserted. The exponential-decay and positive-speed theorems therefore require an additional hypothesis such as E[deg(ρ)S(ρ)]<∞ (or E[deg³(ρ)]<∞), or a separate argument that avoids the stationary mean of S being finite.
  4. [§4.3, full-support proof] In the argument that supp(μ)=∂D, the proof chooses an arc (θ,ψ) disjoint from supp μ, defines A as the vertices whose hyperbolic centers lie in the sector S(θ,ψ), and then asserts ‘Since circles accumulate at the boundary inside this sector, there are infinitely many such vertices u’. This is precisely the kind of statement that needs proof: supp μ being a proper subset of ∂D does not by itself imply that the circle centers accumulate at every point of that arc. If the packing has a boundary gap in that sector, the set A may be finite and the mass-transport contradiction does not follow. The step needs either a proof that the accumulation set of the ICP has no open gaps, or a different argument for full support.
  5. [§4.1, Theorem 4.3 (Refined Ring Lemma)] The proof of the refined ring lemma is too compressed. Lemma 4.2 is stated with a one-line proof, and Lemma 4.4’s iterative contradiction is under-bookkept, particularly the passage ‘This process continues until … which contradicts Lemma 2’. Since Theorem 4.3 is the bridge transferring exponential decay from the induced walk to the original walk in Eq. (26), and is also used in Theorem 5.1 via Eq. (36), this estimate is load-bearing for the boundary and speed theorems. The exponential dependence on S(u) — the main place where unbounded degrees are controlled — needs a complete and self-contained proof.
minor comments (5)
  1. [§3.2] Typo ‘ince’ in ‘ince these angles are independent’. More substantially, the notation IAG+ in Definition 2.4 is introduced without a separate displayed definition of the space IAG+.
  2. [§2.1, Definition 2.1 and 2.4] The angle condition is written ‘Θ2 (0,π )E’ and ‘Θ2 (0,π−ε]E’ with missing superscripts; these should be Θ ∈ (0,π)^E and Θ ∈ (0,π−ε]^E.
  3. [§5] The notation E♮ in Eq. (37) is used before the reversible measure has been consistently named; elsewhere it is E_rev. Please unify the notation.
  4. [§1.3, Remark after Theorem 1.5] The Gromov/Martin boundary identification relies on the unpublished/preparation reference [26]. If these statements are part of the main contribution, they should either be stated as assumptions in the theorems or proved here; otherwise the dependence should be clearly flagged as external.
  5. [Abstract] The abstract states the character dichotomy without the hypotheses of Theorem 1.2 (tameness, ergodicity, E[deg(ρ)]<∞). It would help readers to include these qualifiers in the abstract.

Circularity Check

1 steps flagged · score 4.0 of 10

Proof of Theorem 1.2 invokes 'The Dichotomy Theorem 1.2'—the very theorem being proved—as its first premise.

  1. other [Section 3.2, Proof of Theorem 1.2]
    "First, by Theorem 1.1, we have E[T (ρ)] = E[θ(ρ)]. Then, by The Dichotomy Theorem 1.2, we know E[T (ρ)]≥ 2π, and E[T (ρ)] = 2π if and only if (G,ρ ) has average curvature zero . Thus, we know that statements (1), (2), (3) are equivalent."

    Theorem 1.2 is the dichotomy theorem being proved in this section, yet its proof begins by assuming the theorem's own conclusions ('E[T(ρ)]≥2π', 'E[T(ρ)]=2π iff average curvature zero', equivalence of statements (1)-(3)) 'by The Dichotomy Theorem 1.2' with no external citation. If '1.2' was intended to refer to an external AHNR theorem, the sentence should cite [4,5]; as written, the proof reduces the dichotomy to an assertion of the dichotomy. The later mass-transport argument does provide independent evidence for the ICP-parabolic/hyperbolic directions, so the circularity is localized to the written proof structure rather than necessarily invalidating the whole theorem.

full rationale

The central mass-transport identity E[T(ρ)] = E[θ(ρ)] (Theorem 1.1) is derived honestly from the Mass Transport Principle, and the claimed dichotomy is explicitly framed as an analogue of the external Angel-Hutchcroft-Nachmias-Ray dichotomy, so the result has independent mathematical content. However, as written, Section 3.2's proof of Theorem 1.2 opens by invoking 'The Dichotomy Theorem 1.2'—the theorem under proof—to establish the key inequalities and equivalences. This is a self-referential step in the proof as printed. The same phrase reappears later in the proof to infer invariant non-amenability and VEL-hyperbolicity from ICP-hyperbolicity, compounding the issue. I do not count the reliance on the same authors' separate rigidity/existence results [27] as circular, because those are distinct geometric theorems about existence and uniqueness of circle packings, not the target dichotomy. The skeptical concern about a missing third-moment integrability condition in Lemma 4.5 is a correctness gap, not a circularity: it does not make an output equal to an input by construction. Overall, the written proof contains a load-bearing self-reference, but the theorem's central claim still rests on an external benchmark and an independent mass-transport computation, so a moderate score of 4 is appropriate.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's inputs are (a) published external results ([3,4,5,37,44]), (b) same-group but unverifiable items ([23,27] preprints, [40] unpublished, [26] in preparation) supplying the ICP/IHP foundations and the character, and (c) in-paper estimates (refined ring lemma, mass transports). No fitted parameters exist: ε_0, ε_1 are structural hypotheses and the C(ε_0,ε_1) constants are quantified, not tuned. Note that the angle-pinching Θ ∈ (0,π−ε_1] together with (C1) forces every face to have at most 2π/ε_1 sides, bounding face sizes; it does not bound vertex degrees, which is where the third-moment gap in Lemma 4.5/Thm 5.1 bites. The most serious ledger entry is the implicit moment condition on S under the stationary measure — an unflagged strengthening of the stated hypotheses.

assumptions (7)
  • standard math Rivin's characterization (Thm 1.0, [44]): finite cellular decompositions of S² satisfying (C1),(C2) are precisely the duals of ideal hyperbolic polyhedra with prescribed dihedral angles, unique up to isometry.
    Imported published result that legitimizes defining IAG via the angle conditions; frames the polyhedron–circle-packing correspondence.
  • domain assumption ICP existence and rigidity for infinite IAG (Ge-Yu-Zhou [27, Thm 1.1, 1.7]; Ge-Hua-Yu-Zhou [23]): every tame IAG with Θ ∈ (0,π−ε] admits an embedded ICP/IHP; parabolic and hyperbolic embeddings are unique up to affine/Möbius maps.
    Load-bearing for the well-definedness of ICP-parabolic/hyperbolic type and the IHP/IAG duality used throughout. Cited to same-group preprints (arXiv:2506.19528, arXiv:2506.05036) not independently verified here.
  • standard math AHNR dichotomy and machinery ([4],[5]): for unimodular random planar maps, E[θ(ρ)] ≥ 2π with equality iff invariantly amenable iff VEL-parabolic, and the 17-way equivalence list; [4, Thm 3.2] supplies a bounded-degree percolation with positive Cheeger constant in non-amenable unimodular graphs.
    External published benchmark. The amenability/VEL half of Theorem 1.2 and the induced-walk construction in Lemma 4.5 rest on it.
  • standard math ABGN/HP boundary templates ([3],[37]): for bounded-degree planar triangulations, ∂D realizes the Poisson and Martin boundaries; Lemmas 4.11–4.14 and the non-atomicity/full-support lemmas are quoted from [4].
    External published results. The paper's boundary theory is an extension of these; the quoted lemmas are not re-proven.
  • domain assumption Refined ring lemma (Thm 2.9/4.3): for a tame IAG with Θ ∈ [0,π−ε_1], an embedded ICP satisfies r(v)/r(u) > e^{−C·S(u)} with S(u) = Σ_{v'∼u}deg(v').
    Proved in-paper but via compressed steps (Lemmas 4.1, 4.2, 4.4). Every unbounded-degree conclusion (Lemma 4.5, Thm 5.1) depends on it; identified as the weakest assumption.
  • ad hoc to paper Walk-environment ergodicity with finite mean for S: Birkhoff averages (1/n)ΣS(X_i) converge to a finite limit (used at eq. (28) and Thm 5.1).
    Requires E_rev[S(ρ)] = E[deg(ρ)S(ρ)]/E[deg(ρ)] < ∞, a third-moment-type condition not stated among the hypotheses (only E[deg²(ρ)] < ∞ is assumed). The text conflates the unimodular identity E[S] = E[deg²] with the stationary limit.
  • ad hoc to paper Single-IAG Gromov/Martin boundary identification ([26], in preparation): v ↦ z(v) is a coarse quasi-isometry, ∂G_G = ∂D, and under bounded degree ∂M_G = ∂D with the Martin kernel representation.
    Cited in the Remark after Theorem 1.5 as the basis for the random-setting Gromov/Martin boundary claims; 'in preparation' means it cannot be checked by a reader.

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Pith. "Pith review of Random infinite ideal angled graphs and ideal hyperbolic polyhedra." pith.science (2026). https://pith.science/paper/RH4HWEGR

@misc{pith2026260114909,
  author       = {Pith},
  title        = {Pith review of: Random infinite ideal angled graphs and ideal hyperbolic polyhedra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RH4HWEGR}},
  note         = {Machine review of arXiv:2601.14909}
}
abstract

This article aims to develop the theory of random infinite ideal hyperbolic polyhedra (abbr. IHP) from multiple perspectives, including combinatorics, geometry, analysis, and random walks. Our starting point is the one-to-one correspondence between IHP and ideal circle packings (ICP), which allows us to translate the theory of IHP into the language of ICP. We then extend the theories of Angel-Hutchcroft-Nachmias-Ray \cite{AHNR16,map} to the ICP setting. This extension is far from straightforward: the presence of dihedral angles introduces substantial new difficulties, requiring new estimates, techniques, and theoretical tools. In particular, we introduce a geometric characteristic number that provides a precise and effective characterization of infinite hyperbolic polyhedra. An IHP $\mathcal P$ corresponds to a weighted planar infinite graph $(G,\Theta)$, called an ideal angled graph (abbr. IAG). For unimodular random IAG, we establish an ICP analog of the dichotomy theorem of Angel-Hutchcroft-Nachmias-Ray \cite{AHNR16,map}. Specifically, the geometric characteristic number $T(\rho)=2\pi-\sum_{e\ni\rho}\Theta_e$ of an IAG determines its ICP type: the graph is a.s. ICP-parabolic iff $\mathbb{E}[T(\rho)]=0$. In the ICP-hyperbolic case, the simple random walk converges a.s. to $\partial\mathbb{D}$ with positive hyperbolic speed. Moreover, the geometric, Poisson, Martin boundaries coincide, extending the boundary theory of Angel-Barlow-Gurevich-Nachmias \cite{ABGN16} and Hutchcroft-Peres \cite{HP17} beyond triangulations to cellular decompositions. As a corollary of the aforementioned IHP/IAG duality, we obtain systematic characterizations of random IHPs.

Figures

Figures reproduced from arXiv: 2601.14909 by the authors.

Figure 1
Figure 1. ICP-IHP correspondence and IHP/IAG duality. Rivin’s work sits in a broader picture relating polyhedra in H3 to circle packings on the sphere. In his famous book [54, Chapter 13], Thurston interpreted Andreev’s characterization of hyperbolic polyhedra as a theorem (i.e. the Koebe-Andreev-Thurston theorem) about circle packings. Roughly speaking, circle packings are arrangements of circles on the plane, where the circ… view at source ↗
Figure 2
Figure 2. Ideal circle packing The angle Θ : E → (0, π) may satisfy the Rivin-Thurston combinatorial conditions [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Quadrilateral We write α(e,v) = α w v ∈ (0, 2π) for the angle ∠vf1 vvf2 in Q˜ e (see [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Mass transport m(u, v) P Now, we compute the total mass sent from a fixed vertex. Fix u ∈ V (G). We compute v∈V (G) m(u, v). Starting from (6), since M is locally finite and each face has finite degree, for fixed u only finitely many terms in the sums are nonzero; henc…
Figure 5
Figure 5. Figure 5: Mass transport between ρ and v2 and (11) X v∼ρ F(G, v, ρ) = F(G, ρ, ρ) + X v∼ρ,v̸=ρ F(G, v, ρ) = 2π + Xn i=1 ∠vfi vivfi+1 = Xn i=1 ∠vfi ρvfi+1 + Xn i=1 ∠vfi vivfi+1 = 2Xn i=1 (θ (fi) − π + Θρvi ) = 2T(ρ) + 2Xn i=1 (θ (fi) − π), where θ (fi) is the sum of the internal a…
Figure 6
Figure 6. Figure 6: Graph for Lemma 4.1 We have P i (π − Θ′ i ) = 2π. For each i, the following inequality holds [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: The orange arc must be covered u v1 v2 a b w1 w2 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: A possible arrangement of circles Proof. Let E be the union of disks v1, v2, . . . , vn, and w1, w2, . . . , wm be disks around b (except disks in E and the center u), Then diam(E) < Pn i=1 rvi . if a and b intersect: we are done with (C ′ 2 ) and C3 = C(ε0) from Lemma…
Figure 9
Figure 9. Figure 9: The center circle can be arbitrarily small when ε0 tends to 0 4.2. Convergence. The focus of this subsection is to establish the convergence and to analyze the resulting exit distribution, thereby proving items (1) and (2) of Theorem 1.4. Lemma 4.5 (Exponential decay o…
Figure 10
Figure 10. Figure 10: The vertex v receives mass from circles with hyperbolic centres in the shaded area from e iϕ to e iθ. For a vertex v whose circle intersects γ, define Bv := n ϕ ∈ (θ, ψ) : v is the first circle met by γϕ that also intersects γ o . Up to a null set of exceptional ϕ whe…

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Works this paper leans on

54 extracted references · 4 linked inside Pith

  1. [5]

    Omer Angel, Tom Hutchcroft, Asaf Nachmias and Gourab Ray, Hyperbolic and parabolic unimodular random maps. Geom. Funct. Anal. 28 (2018), 879-942

    2018

  2. [26]

    Gromov and Martin boundaries of ideal angled graphs, in preparation

    Huabin Ge, Yangxiang Lu, Chuwen Wang, Tian Zhou. Gromov and Martin boundaries of ideal angled graphs, in preparation

  3. [1]

    Electron

    David Aldous and Russell Lyons, Processes on unimodular random networks. Electron. J. Probab. 12 (2007), Paper 54, 1454-1508

    2007

  4. [2]

    E. M. Andreev, Convex polyhedra in Lobačevskiĭ spaces. Math. USSR Sb. 10 (1970), 413-440

    1970

  5. [3]

    Barlow, Ori Gurel-Gurevich and Asaf Nachmias, Boundaries of planar graphs, via circle packings

    Omer Angel, Martin T. Barlow, Ori Gurel-Gurevich and Asaf Nachmias, Boundaries of planar graphs, via circle packings. Ann. Probab. 44 (2016), no. 3, 1956-1984

    2016

  6. [4]

    Omer Angel, Tom Hutchcroft, Asaf Nachmias and Gourab Ray, Unimodular hyperbolic triangulations: circle packing and random walk. Invent. Math. 206 (2016), no. 1, 229-268

    2016

  7. [6]

    Omer Angel and Oded Schramm, Uniform infinite planar triangulations. Comm. Math. Phys. 241 (2003), no. 2-3, 191-213

    2003

  8. [7]

    Xiliang Bao and Francis Bonahon, Hyperideal polyhedra in hyperbolic 3-space. Bull. Soc. Math. France 130 (2002), no. 3, 457-491

    2002

Show all 54 references
  1. [8]

    Itai Benjamini and Oded Schramm, Harmonic functions on planar and almost planar graphs and manifolds via circle packings. Invent. Math. 126 (1996), no. 3, 565-587

    1996

  2. [9]

    Itai Benjamini and Oded Schramm, Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 (1996), no. 3, 1219-1238

    1996

  3. [10]

    Itai Benjamini and Oded Schramm, Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal. 7 (1997), no. 3, 403-419

    1997

  4. [11]

    Electron

    Itai Benjamini and Oded Schramm, Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001), no. 23, 1-13

    2001

  5. [12]

    Itai Benjamini and Nicolas Curien, Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points. Geom. Funct. Anal. 23 (2013), no. 2, 501-531

    2013

  6. [13]

    Preprint, 2014

    Itai Benjamini, Elliot Paquette and Joshua Pfeffer, Anchored expansion, speed, and the hyperbolic Poisson Voronoi tessellation. Preprint, 2014. arXiv:1404.4685

    2014 arXiv

  7. [14]

    Olivier Bernardi, Nathanaël Holden, and Xin Sun, Percolation on triangulations: a bijective path to Li- ouville quantum gravity. Mem. Amer. Math. Soc. 289 (2023), no. 1440, American Mathematical Society, Providence, RI

    2023

  8. [15]

    Bobenko and Boris A

    Alexander I. Bobenko and Boris A. Springborn, Variational principles for circle patterns and Koebe’s theorem. Trans. Amer. Math. Soc. 356 (2004), no. 2, 659-689

    2004

  9. [16]

    American Mathematical Society, Providence, RI, 2001

    Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. 46

    2001

  10. [17]

    H. S. M. Coxeter, Self-dual configurations and regular graphs. Bull. Amer. Math. Soc., 56:413-455, 1950

    1950

  11. [18]

    Nicolas Curien, Planar stochastic hyperbolic triangulations. Probab. Theory Relat. Fields (2015), 1-32

    2015

  12. [19]

    Nicolas Curien and Jean-François Le Gall, The Brownian plane. J. Theoret. Probab. 27 (2014), no. 4, 1249-1291

    2014

  13. [20]

    Nicolas Curien and Jean-François Le Gall, The hull process of the Brownian plane. Probab. Theory Related Fields 166 (2016), no. 1-2, 187-231

    2016

  14. [21]

    ALEA Lat

    Nicolas Curien, Laurent Ménard and Grégory Miermont, A view from infinity of the uniform infinite planar quadrangulation. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 45-88

    2013

  15. [22]

    Jian Ding and Ewain Gwynne, The fractal dimension of Liouville quantum gravity: universality, mono- tonicity, and bounds. Comm. Math. Phys. 374 (2020), 1877-1934

    2020

  16. [23]

    Preprint, 2025

    Huabin Ge, Bobo Hua, Hao Yu and Puchun Zhou, Characterization of infinite ideal polyhedra in hyperbolic 3-space via combinatorial Ricci flow. Preprint, 2025. arXiv:2506.05036

    2025 arXiv

  17. [24]

    Huabin Ge, Bobo Hua and Ze Zhou, Circle patterns on surfaces of finite topological type. Amer. J. Math. 143 (2021), no. 5, 1397-1430

    2021

  18. [25]

    Huabin Ge, Bobo Hua and Ze Zhou, Combinatorial Ricci flows for ideal circle patterns. Adv. Math. 383 (2021), Paper No. 107698, 26 pp

    2021

  19. [27]

    Preprint, 2025

    Huabin Ge, Hao Yu and Puchun Zhou, Infinite ideal polyhedra in hyperbolic 3-space: existence and rigidity. Preprint, 2025. arXiv:2506.19528

    2025 arXiv

  20. [28]

    Agelos Georgakopoulos, The boundary of a square tiling of a graph coincides with the Poisson boundary. Invent. Math. 203 (2016), no. 3, 773-821

    2016

  21. [29]

    Ori Gurel-Gurevich and Asaf Nachmias, Recurrence of planar graph limits. Ann. Math. (2) 177 (2013), no. 2, 761-781

    2013

  22. [30]

    Ewain Gwynne and Jason Miller, Random walk on random planar maps: Spectral dimension, resistance, and displacement. Ann. Probab. 49 (2021), no. 3, 1097-1128

    2021

  23. [31]

    Preprint (2019), arXiv:1910.04713

    Ewain Gwynne, Nina Holden and Xin Sun, Mating of trees for random planar maps and Liouville quantum gravity: A survey. Preprint (2019), arXiv:1910.04713

    2019 arXiv

  24. [32]

    Zhengxu He, On the convergence of circle packings to the Riemann map. Invent. Math. 125 (1996), no. 2, 285-305

    1996

  25. [33]

    Zhengxu He, Rigidity of infinite disk patterns. Ann. Math. (2) 149 (1999), no. 1, 1–33

    1999

  26. [34]

    Zhengxu He and Oded Schramm, Fixed points, Koebe uniformization and circle packings. Ann. Math. (2) 137 (1993), no. 2, 369-406

    1993

  27. [35]

    Discrete Comput

    Zhengxu He and Oded Schramm, Hyperbolic and parabolic packings. Discrete Comput. Geom. 14 (1995), no. 2, 123-149

    1995

  28. [36]

    Acta Math

    Nathanaël Holden and Xin Sun, Convergence of uniform triangulations under the Cardy embedding. Acta Math. 230 (2023), 93-203

    2023

  29. [37]

    Electron

    Tom Hutchcroft and Yuval Peres, Boundaries of planar graphs: a unified approach. Electron. J. Probab. 22 (2017), Paper 100, 1-20

    2017

  30. [38]

    Harry Kesten, Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 (1959), 336-354

    1959

  31. [39]

    Paul Koebe, Kontaktprobleme der konformen Abbildung. Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.- Phys. Kl. 88 (1936), 141-164

    1936

  32. [40]

    Unpublished manuscript (2025)

    Chang Li, Aijin Lin, and Liangming Shen, The character of ideal circle patterns. Unpublished manuscript (2025)

    2025

  33. [41]

    Cambridge Univ

    Russell Lyons and Yuval Peres, Probability on trees and networks . Cambridge Univ. Press, New York, 2016

    2016

  34. [42]

    Topology 32 (1993), no

    Igor Rivin, On geometry of convex ideal polyhedra in hyperbolic 3-space. Topology 32 (1993), no. 1, 87-92

    1993

  35. [43]

    Igor Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. (2) 139 (1994), no. 3, 553-580

    1994

  36. [44]

    Igor Rivin, A characterization of ideal polyhedra in hyperbolic 3-space. Ann. Math. (2) 143 (1996), no. 1, 51-70

    1996

  37. [45]

    Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping. J. Differential Geom. 26 (1987), no. 2, 349-360

    1987

  38. [46]

    Steffen Rohde, Oded Schramm: from circle packing to SLE. Ann. Probab. 39 (2011), 1621-1667

    2011

  39. [47]

    Jean-Marc Schlenker, Hyperideal circle patterns. Math. Res. Lett. 12 (2005), no. 1, 85-112

    2005

  40. [48]

    Discrete Comput

    Jean-Marc Schlenker, Circle patterns on singular surfaces. Discrete Comput. Geom. 40 (2008), no. 1, 47-102. 47

    2008

  41. [49]

    Jean-Marc Schlenker, Hyperideal polyhedra in hyperbolic manifolds. Math. Res. Lett. 20 (2013), no. 4, 773-786

    2013

  42. [50]

    Israel J

    Oded Schramm, Square tilings with prescribed combinatorics. Israel J. Math. 84 (1993), no. 1-2, 97-118

    1993

  43. [51]

    Duke Math

    Oded Schramm, Circle patterns with the combinatorics of the square grid. Duke Math. J. 86 (1997), no. 2, 347-389

    1997

  44. [52]

    Discrete Comput

    Boris Springborn, Ideal hyperbolic polyhedra and discrete uniformization. Discrete Comput. Geom. 64 (2020), no. 1, 63-108

    2020

  45. [53]

    Cambridge Univ

    Kenneth Stephenson, Introduction to circle packing: the theory of discrete analytic functions . Cambridge Univ. Press, Cambridge, 2005

    2005

  46. [54]

    Thurston, The geometry and topology of three-manifolds

    William P. Thurston, The geometry and topology of three-manifolds . Princeton Univ. lecture notes, 1978– 1981, available at Thurston’s webpage. Author information Huabin Ge Renmin University of China E-mail: hbge@ruc.edu.cn Chuwen W ang Renmin University of China E-mail: chuwe...

    1978

Pith tools

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