Pith. sign in

REVIEW 5 major objections 5 minor 54 references

A Gauss-Bonnet-Type Dichotomy for Unimodular Random Infinite Trivalent Hyperbolic Polyhedra

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that for tame, ergodic, unimodular random trivalent hyperbolic polyhedra with finite expected face degree, the global conformal type is parabolic exactly when the expected facewise angle defect $\mathbb{E}[L_f(P)]$ is…

desk verdict The local Gauss–Bonnet formula and the soficity obstruction are solid and worth publishing; the advertised global dichotomy is a bridge over two unpublished preprints and needs external verification before it can be taken at face value. read the letter →

arxiv 2608.03575 v1 pith:QZESOGQQ submitted 2026-08-04 math.PR math.DGmath.GT

Huabin Ge , Yangxiang Lu , Chuwen Wang , Tian Zhou This is my paper · ORCID
classification math.PRmath.DGmath.GT MSC 60D0552C26
keywords unimodularrandomgraphstrivalenthyperbolicpolyhedracirclepatternsGauss-BonnetformulaconformaltypePoissonboundaryringlemmaBenjamini-Schrammlimits
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 tries to establish a local-to-global principle: for an infinite random hyperbolic polyhedron built as the intersection of half-spaces in $\mathbb{H}^3$, the global conformal type—parabolic, where the truncated polyhedron escapes to a single point on the sphere at infinity, or hyperbolic, where it escapes to a circle—is determined by the expectation of a single local quantity attached to the root face. That quantity, the geometric characteristic number $L_f(P) = 2\pi - \sum_{v\in f}\theta_v^f$, is a facewise angle defect computed from the dihedral angles along the edges meeting each vertex. The paper proves the unimodular Gauss-Bonnet formula $\mathbb{E}[L_f(P)] = 2\pi - (\pi/3)\mathbb{E}[\deg(f)]$, so the sign of the expectation is equivalent to whether the expected face degree is $6$ or larger. Under the paper's tameness and admissibility conditions this yields the dichotomy $\mathbb{E}[L_f(P)]=0$ exactly in the parabolic case and $\mathbb{E}[L_f(P)]<0$ exactly in the hyperbolic case, and also shows that hyperbolic examples cannot be approximated by finite trivalent hyperbolic polyhedra. If the argument is right, a statistician or geometer who can estimate the expected degree of one face knows the conformal type, boundary behaviour, and escape rate of the entire random polyhedron.

What carries the argument

The load-bearing object is the geometric characteristic number $$L_f(P)=2\pi-\sum_{v\in f}\theta_v^f,$$ where $\theta_v^f$ is the angle at vertex $v$ in an auxiliary Euclidean triangle whose side lengths come from the three dihedral angles at that trivalent vertex; the defining identity is that the three angles at each vertex sum to $\pi$. The argument's mechanism is the mass transport principle applied to a specifically chosen transport on the dual angled disk triangulation: the mass sent from a root face is $6\pi-3L_f(P)$, the mass received is $\pi\deg(f)$, and unimodularity equates the expectations, giving the Gauss-Bonnet identity. For the stochastic half, the key tool is the refined ring lemma, an exponential radius-ratio bound $r(v)/r(u)>e^{-C S(u)}$ where $S(u)=\sum_{w\sim u}\deg(w)$ is the flower degree of $u$; this replaces the uniform bounds of classical ring lemmas in the presence of unbounded degrees and supplies the integrability needed for boundary convergence and positive speed.

What would settle it

Sample an explicit ergodic unimodular random angled disk triangulation (for instance, a random disk triangulation with admissible i.i.d. angle marks), solve its regular circle pattern numerically, and compare root expected degree with the computed VEL or circle-packing type: a realization with $\mathbb{E}[\deg(\rho)]=6$ that is RCP-hyperbolic, or with $\mathbb{E}[\deg(\rho)]>6$ that is RCP-parabolic, would falsify Theorem 3.2 and the dichotomy it drives.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a three-way rigidity: local dihedral geometry, global conformal type, and stochastic asymptotics are locked together by one expectation. Theorem 1.4 states that for a tame, infinite, ergodic, unimodular random trivalent hyperbolic polyhedron satisfying (Z3) and (Z4) with $\mathbb{E}[\deg(f)]<\infty$, the root-face characteristic satisfies $\mathbb{E}[L_f(P)]=0$ if and only if the polyhedron is almost surely parabolic, and $\mathbb{E}[L_f(P)]<0$ if and only if it is almost surely hyperbolic. The engine is the mass transport identity $\mathbb{E}[L_f(P)] = 2\pi - (\pi/3)\mathbb{E}[\deg(f)]$, whose angular dependence cancels because the three auxiliary Euclidean angles at each trivalent vertex sum to $\pi$. The paper further proves that every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is parabolic, so genuinely hyperbolic structures are not polyhedrally sofic; and, under a strengthened cycle condition and a third-moment assumption, that for hyperbolic examples the face random walk converges to the circle at infinity, that the exit measure is non-atomic with full support and realizes the Poisson boundary, and that the walk has positive linear speed.

Load-bearing premise

The load-bearing premise is that every simple closed curve in the dual triangulation that is not the boundary of a face has total angle at most $(s-2)\pi - \varepsilon_0$ for one fixed deterministic $\varepsilon_0>0$, and the paper gives no argument that a typical unimodular random trivalent hyperbolic polyhedron satisfies this strengthened cycle condition.

Editorial extensions

If this is right

  • The conformal type of the whole random polyhedron is controlled by one local observable: the expected geometric characteristic of the root face, with the angle configuration otherwise cancelling in expectation.
  • The degree threshold is universal: expected face degree $6$ forces parabolicity, expected degree $>6$ forces hyperbolicity, independently of the numerical values of the dihedral angles (within the admissibility conditions).
  • Every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is parabolic, and every hyperbolic unimodular trivalent hyperbolic polyhedron satisfying the standing assumptions is not polyhedrally sofic.
  • In the hyperbolic regime, under the strengthened cycle condition and finite third moment, the face random walk converges almost surely to a point of the circle at infinity, the exit measure is atom-free with full support and realizes the Poisson boundary, and the walk escapes with positive linear speed.
  • On trivalent ideal polyhedra the new invariant agrees with the previously studied angle defect, so the dichotomy extends the ideal theory to ordinary, ideal, and hyperideal vertices without changing the answer.

Reading between the lines

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

  • A consequence the paper leaves implicit is that the classification is insensitive to the angle distribution itself: any two admissible angle laws on the same random combinatorial model should give the same conformal type whenever they share the same expected degree.
  • The strengthened cycle condition is the bottleneck for making the stochastic conclusions unconditional; proving it almost surely for natural random polyhedral models (for instance, limits of finite polyhedra with compactly supported angle data) would be the direct next step.
  • The no-hyperbolic-limit theorem is special to spherical approximations; the paper's own remarks indicate that approximating the same graph by higher-genus surfaces bypasses the finite-spherical Gauss-Bonnet obstruction, so polyhedral soficity and ordinary graph soficity are plausibly different properties.
  • The refined ring lemma's exponential control in flower degree is a general analytic tool; it should transfer to other angle-prescribed discrete conformal structures (square tilings, hyperideal circle patterns), carrying the Poisson-boundary and positive-speed results with it.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 probabilistic geometric theory for unimodular random infinite trivalent hyperbolic polyhedra. It defines a local face characteristic L_f(P) from dihedral angles and proves the unimodular Gauss-Bonnet identity E[L_f(P)] = 2π - (π/3) E[deg(f)] by a mass transport argument. The sign of this expectation is then claimed to determine the conformal type: zero expectation corresponds to parabolicity and negative expectation to hyperbolicity, with equivalent characterizations via vertex extremal length, regular circle pattern type, and invariant amenability. The paper also proves that Benjamini-Schramm limits of uniformly face-rooted finite trivalent hyperbolic polyhedra are parabolic, establishes a refined ring lemma for regular circle patterns under a strengthened angle condition (Z2'), and uses it to identify the Poisson boundary with the circle at infinity and to prove positive speed for the face random walk.

Significance. The claimed dichotomy is attractive and potentially important: it would reduce the global conformal type of a unimodular random hyperbolic polyhedron to the expectation of a purely local angle defect. The proof of Theorem 3.1 is a clean, fully written mass transport argument, and the finite Gauss-Bonnet formula in Proposition 4.2 together with the Benjamini-Schramm parabolicity theorem is a genuinely interesting and non-obvious obstruction. The refined ring lemma is a genuinely useful quantitative tool for unbounded-degree circle patterns. The main caveat is that the global dichotomy is not self-contained: its decisive equivalences are delegated to external preprints and papers without reproducing the hypotheses or conclusions, and several Section 6 arguments are too terse to verify as written.

major comments (5)
  1. [§3, Theorem 3.2] The central equivalence (iii)↔(iv) in Theorem 3.2 is asserted by citing [26, Theorem 1.4], but that theorem is neither stated nor checked against the present hypotheses. The paper works with tame ADTs satisfying (Z1)–(Z4), allows unbounded degrees, only assumes E[deg(ρ)]<∞, admits weak RCP realizations, and allows angle values in [0,π−ε]. If [26, Theorem 1.4] requires bounded degree, strictly positive angles, or exact realizations, then the equivalence fails precisely in the advertised regime. This dependency is load-bearing because it is what turns the local Gauss-Bonnet identity into the global conformal dichotomy. The authors should either state the exact theorem and verify its hypotheses, or prove the needed equivalence directly.
  2. [§2.2, Definition 2.11 and following paragraph] The paper defines RCP-parabolic/RCP-hyperbolic by existence of a realization with carrier C or D, but does not prove that this type is well-defined under the standing assumptions, nor that it agrees with VEL type for tame ADTs with unbounded degrees and weak realizations. The assertion 'this analytic type agrees with the VEL type' is made without proof. Since this agreement is used in the proof of Theorem 1.4 and in Section 6, it must either be proved or precisely attributed with a verifiable statement. As written, the reader cannot determine what assumptions the cited uniformization theorem needs.
  3. [§5, Lemma 5.1 and Definition 2.2] The stochastic half of the paper, including the refined ring lemma and the entire boundary theory in Section 6, depends on the strengthened cycle condition (Z2'), but the paper gives no argument that a typical unimodular random tame THP or ADT satisfies a uniform margin ε0>0. Without such an argument, Lemma 5.1, Corollary 1.6, and Corollary 1.7 are conditional on a hypothesis that is not established for the main class. In addition, Section 6 assumes E[deg(ρ)^3]<∞, which is stronger than the E[deg(ρ)]<∞ used in the dichotomy; the paper should clarify how these assumptions are to be satisfied in intended applications.
  4. [§6.3, Lemmas 6.14 and 6.15] The proof of non-atomicity in Lemma 6.14 relies on the assertion that the straight-line realization obtained from the RCP is a planar embedding compatible with the underlying triangulation; this is not proved and is not immediate for weak realizations or unbounded-degree patterns. The proof of full support in Lemma 6.15 is also extremely compressed: the 'standard planar mass-transport argument' is not described, and it is not clear why a boundary component omitted from the support forces a separator to receive infinite mass. These two lemmas are essential for the Poisson boundary identification, so the arguments need to be written out in full.
  5. [§3, proof of Theorem 1.4] The proof of Theorem 1.4 says that 'under the THP/ADT correspondence, the parabolic/hyperbolic type of P agrees with the RCP type of (G,Θ)', but this correspondence is not established in the manuscript. Theorem 2.10 constructs a THP from an RCP and identifies dihedral angles, but it does not prove that the parabolic/hyperbolic type of the resulting polyhedron matches the RCP carrier type under the present hypotheses (which include ordinary, ideal, and hyperideal vertices). This is a second, independent gap in the chain from the local Gauss-Bonnet formula to the THP dichotomy.
minor comments (5)
  1. [§2.1, Definition 2.2] The wording of (Z3) is confusing: 'homologically non-adjacent edges' is defined through an arc formed by two adjacent edges, but the phrase 'homologically adjacent' suggests a homology condition rather than a combinatorial one. Please rephrase or add a figure.
  2. [§5, Lemma 5.3] In Lemma 5.3, the set E_u(x) is defined as a union of D_v\D_u, but the proof later uses points P_xy and P_xz that lie on boundaries of neighboring disks; the inclusion of these points in E_u(x) should be stated explicitly.
  3. [§5.4 (Lemma 5.4)] The phrase 'if it lies inside the disk D_a, the intersection graph of centers a,b,y,u would form a planar K4 with b in the center' is not clear; this step needs a more detailed geometric explanation.
  4. [§6.3, Lemma 6.12] The compatibility of the family of metrics {d_G^{(u)}} is asserted but the 'remaining rotational ambiguity' is not fully discussed. Since the boundary identification depends on this compatibility, a few sentences of justification would help.
  5. [Throughout] There are several small formatting issues, including 'P^{trun}' in Definition 2.11 and the unlabeled equation (2) referenced in the proof of Theorem 3.1; please ensure all displayed equations are numbered consistently.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the unimodular Gauss-Bonnet identity is proved by mass transport, and the dichotomy imports external type-equivalence theorems rather than assuming its own conclusion.

full rationale

The central derivation in Theorem 3.1 is self-contained. It defines a mass transport function m(u,v), computes the total mass sent from the root as 6π - 3Lρ(G,Θ) and the total mass received as π degG(ρ), and applies the mass transport principle to obtain E[Lρ(G,Θ)] = 2π - (π/3)E[degG(ρ)]. This is a genuine identity, not a fitted parameter or a renamed prediction. The dichotomy Theorem 3.2 then combines this identity with external results: the equivalence between E[deg(ρ)] = 6 and VEL/amenability is quoted from [8], and the equivalence between VEL type and RCP/THP type is quoted from [26]. These are load-bearing citations, and [24, 26] are preprints by overlapping authors, so there is a legitimate self-citation concern. However, under the stated review rules, a cited theorem with its own stated assumptions that does not include the target result counts as independent support; the paper's conclusion is not used as an input to these cited theorems. The paper also explicitly acknowledges that the identity 'recovers the usual degree-six threshold from the unweighted case', so there is no concealment or renaming of a known result as a new prediction. The soficity obstruction (Theorem 4.3) and the boundary/positive-speed results (Section 6) are proved from the established estimates and do not reduce to the dichotomy statement. No exhibited equation or fitted parameter makes the claimed predictions equivalent to their inputs by construction. The main reliability caveat is that the crucial VEL/RCP/THP type agreement is external and not reproduced, which is a correctness risk rather than circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on the mass transport principle, the Angel-Hutchcroft-Nachmias-Ray dichotomy for unimodular triangulations, and the existence and rigidity theory for regular circle patterns from the authors' earlier preprint [26]. The admissibility conditions (Z1)-(Z4) and the strengthened (Z2') are assumed, not derived; no argument shows they hold for generic random THPs. No free parameters are fitted and no new physical entities are postulated.

assumptions (9)
  • standard math Mass transport principle for unimodular random rooted graphs
    Used in Theorem 3.1 to equate expected mass sent with expected mass received, producing the Gauss-Bonnet formula.
  • standard math Dichotomy theorem for one-ended unimodular random triangulations (Angel-Hutchcroft-Nachmias-Ray [8])
    Gives equivalence of E[deg]=6 with VEL-parabolic and E[deg]>6 with VEL-hyperbolic, used in Theorem 3.2.
  • standard math Existence and rigidity of regular circle patterns realizing an ADT under (Z1)-(Z4) (Theorems 2.4 and 2.5 in [26])
    Bridges ADTs to RCPs and THPs; cited from the authors' own preprint.
  • standard math Agreement of VEL type, RCP type, and THP parabolic/hyperbolic type under the angle conditions ([26, Theorem 1.4])
    Transfers the ADT dichotomy to THPs in Theorem 1.4.
  • standard math Counting lemma for large circles in circle patterns ([24, Lemma 4.8])
    Used in Lemma 6.8 to bound the number of circles with radius at least t.
  • standard math Existence of an invariant non-amenable bounded-degree subgraph of an invariantly non-amenable graph ([7], [8], [41])
    Provides the positive escape rate and exponential decay of radii in Section 6.
  • domain assumption Tameness: there is a deterministic epsilon > 0 such that all dihedral angles are at most pi - epsilon
    Assumed for all main theorems; prevents degeneracy as angles approach pi.
  • domain assumption Admissibility conditions (Z3) and (Z4) for THPs, and strengthened (Z2') for the ring lemma and boundary results
    Assumed in Theorem 1.4 and Section 6; (Z2') requires a uniform positive gap in the cycle inequality.
  • domain assumption Third moment condition E[deg(rho)^3] < infinity
    Assumed in Corollaries 1.6 and 1.7 to control flower-degree integrability and random walk increments.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Gauss-Bonnet-Type Dichotomy for Unimodular Random Infinite Trivalent Hyperbolic Polyhedra." pith.science (2026). https://pith.science/paper/QZESOGQQ

@misc{pith2026260803575,
  author       = {Pith},
  title        = {Pith review of: A Gauss-Bonnet-Type Dichotomy for Unimodular Random Infinite Trivalent Hyperbolic Polyhedra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZESOGQQ}},
  note         = {Machine review of arXiv:2608.03575}
}
abstract

We develop a unified geometric and probabilistic theory of conformal type for unimodular random infinite trivalent hyperbolic polyhedra in $\mathbb{H}^3$. By corresponding these with dual angled disk triangulations and regular circle patterns, we associate to each face an intrinsic geometric characteristic number $L_f(P)$, determined entirely by local dihedral geometry. For the root face $f$, we establish the unimodular Gauss-Bonnet formula $\mathbb{E}[L_f(P)] = 2\pi - (\pi/3)\mathbb{E}[deg(f)]$. Under natural tameness and admissibility assumptions, this yields a sharp dichotomy: a unimodular random trivalent hyperbolic polyhedron is parabolic precisely when $\mathbb{E}[L_f(P)] = 0$, and hyperbolic when $\mathbb{E}[L_f(P)] < 0$. Thus, global conformal type is governed by the expectation of a local geometric quantity. We also investigate the approximation of infinite polyhedra by finite ones. We prove that every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is necessarily parabolic, revealing a geometric and topological obstruction to the existence of hyperbolic unimodular polyhedral limits. To study stochastic behavior in the hyperbolic regime, we overcome the failure of classical circle packing tools for unbounded degrees by establishing a refined ring lemma for regular circle patterns. This yields effective exponential control of adjacent circle radii via local flower degrees. Combined with boundary methods, we identify the Poisson boundary with the circle at infinity and prove positive hyperbolic speed for the face random walk. These results provide the first quantitative framework connecting local three-dimensional dihedral geometry, global conformal type, and asymptotic stochastic behavior of unimodular random infinite hyperbolic polyhedra.

Figures

Figures reproduced from arXiv: 2608.03575 by the authors.

Figure 1
Figure 1. The mass transport corresponding to adjacent faces X v∈V (G) m(u, v) = X u∼v m(u, v) + 2π − Lu(G, Θ) = X u∼v (θ vw1 u + θ vw2 u ) + 2π − Lu(G, Θ) = 2X f∋u θ f u + 2π − Lu(G, Θ) = 6π − 3Lu(G, Θ), Therefore the mass sent from the root ρ is 6π − 3Lρ(G, Θ). And the mass received: X v∈V (G) m(v, u) = X u∼v m(v, u) + 2π − Lu(G, Θ) = X u∼v (θ uw1 v + θ uw2 v ) + X u∈f θ f u = X f∋u (θ f v + θ f w + θ f u ) (u, v, w are ver… view at source ↗
Figure 2
Figure 2. The polyline colored in blue The final assertion follows from the same cyclic chain. Indeed, if x is chosen so that rx = maxv∼u rv, then the poly-line Pxyyw1 . . . wmzPxz has length l = ry + dyw1 + · · · + dwmz + rz ≤ ry + (ry + rw1 ) + · · · + (rwm + rz) + rz = 2 X v∼u rv − rx ! = 2 X v∼u rv − max v∼u {rv} ! . Since this poly-line connects Pxy to Pxz counterclockwise with respect to u, its length is at least 2ru si… view at source ↗
Figure 3
Figure 3. The case where Dy and Da intersect. following u. These vertices are different from u, a, b, and hence the disks Dx and Dy are both contained in Ea,b(u). For every face rst, denote by Pst the outer intersection point of the two neighboring disks Ds and Dt, as in the proof of Lemma 5.3. By the same local RCP geometry used there, the points Pxu and Puy lie outside the circle Da. Moreover, the sector at a cut out by the… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 53 canonical work pages

  1. [8]

    Angel, T

    O. Angel, T. Hutchcroft, A. Nachmias, and G. Ray,Hyperbolic and parabolic unimodular random maps, Geom. Funct. Anal.28(2018), no. 4, 879–942

    work page 2018

  2. [26]

    H.-B. Ge, L. Jia, H. Yu, and P. Zhou,Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space, arXiv:2511.09368, 2025

    work page Pith review arXiv 2025

  3. [24]

    H.-B. Ge, Y. Lu, C. Wang, and T. Zhou,Random infinite ideal angled graphs and ideal hyperbolic polyhedra, arXiv:2601.14909, 2026

    work page Pith review arXiv 2026

  4. [1]

    D. J. Aldous and R. Lyons,Processes on unimodular random networks, Electron. J. Probab.12(2007), 1454–1508

    work page 2007

  5. [2]

    Angel, M

    O. Angel, M. T. Barlow, O. Gurel-Gurevich, and A. Nachmias,Boundaries of planar graphs, via circle packings, Ann. Probab.44(2016), no. 3, 1956–1984

    work page 2016

  6. [3]

    Angel and O

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

    work page 2003

  7. [4]

    E. M. Andreev,On convex polyhedra in Lobachevskii spaces, Math. USSR-Sb.10(1970), 413–440. 38

    work page 1970

  8. [5]

    E. M. Andreev,On convex polyhedra of finite volume in Lobachevskii space, Math. USSR-Sb.12(1970), 255–259

    work page 1970

Show all 54 references
  1. [6]

    Ancona,Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann

    A. Ancona,Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2) 125(1987), no. 3, 495–536

    1987

  2. [7]

    Angel, T

    O. Angel, T. Hutchcroft, A. Nachmias, and G. Ray,Unimodular hyperbolic triangulations: circle packing and random walk, Invent. Math.206(2016), no. 1, 229–268

    2016

  3. [9]

    Bao and F

    X. Bao and F. Bonahon,Hyperideal polyhedra in hyperbolic 3-space, Bull. Soc. Math. France130(2002), no. 3, 457–491

    2002

  4. [10]

    A. F. Beardon and K. Stephenson,The uniformization theorem for circle packings, Indiana Univ. Math. J.39(1990), no. 4, 1383–1425

    1990

  5. [11]

    A. F. Beardon and K. Stephenson,Circle packings in different geometries, Tohoku Math. J. (2)43(1991), no. 1, 27–36

    1991

  6. [12]

    Benjamini and O

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

    1996

  7. [13]

    Benjamini and O

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

    1996

  8. [14]

    Benjamini and O

    I. Benjamini and O. Schramm,Recurrence of distributional limits of finite planar graphs, Electron. J. Probab.6(2001), no. 23, 1–13

    2001

  9. [15]

    Benjamini and N

    I. Benjamini and N. Curien,Ergodic theory on stationary random graphs, Electron. J. Probab.17(2012), no. 93, 1–20

    2012

  10. [16]

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

    2004

  11. [17]

    Br¨ agger,Kreispackungen und Triangulierungen, Enseign

    W. Br¨ agger,Kreispackungen und Triangulierungen, Enseign. Math. (2)38(1992), no. 3–4, 201–217

    1992

  12. [18]

    R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. R. Tutte,The dissection of rectangles into squares, Duke Math. J.7(1940), no. 1, 312–340

    1940

  13. [19]

    J. W. Cannon, W. J. Floyd, and W. R. Parry,Squaring rectangles: the finite Riemann mapping theorem, in The Mathematical Legacy of Wilhelm Magnus, Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133–212

    1994

  14. [20]

    Colin de Verdi` ere,Un principe variationnel pour les empilements de cercles, Invent

    Y. Colin de Verdi` ere,Un principe variationnel pour les empilements de cercles, Invent. Math.104(1991), no. 3, 655–669

    1991

  15. [21]

    Curien,Random graphs: the local convergence point of view, lecture notes, 2018

    N. Curien,Random graphs: the local convergence point of view, lecture notes, 2018

    2018

  16. [22]

    P. G. Doyle and J. L. Snell,Random Walks and Electric Networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984

    1984

  17. [23]

    Ge and A.-J

    H.-B. Ge and A.-J. Lin,The character of Thurston’s circle packings, Science China Mathematics67(2024), 1623–1640

    2024

  18. [25]

    H.-B. Ge, H. Yu, and P. Zhou,Infinite ideal polyhedra in hyperbolic 3-space: existence and rigidity, arXiv:2506.19528, 2025

    2025 arXiv

  19. [27]

    Georgakopoulos,The boundary of a square tiling of a graph coincides with the Poisson boundary, Invent

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

    2016

  20. [28]

    Gurel-Gurevich and A

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

    2013

  21. [29]

    Gou¨ ezel and S

    S. Gou¨ ezel and S. P. Lalley,Random walks on co-compact Fuchsian groups, Ann. Sci. ´Ec. Norm. Sup´ er. (4)46(2013), no. 1, 129–173

    2013

  22. [30]

    Gou¨ ezel,Martin boundary of random walks with unbounded jumps in hyperbolic groups, Ann

    S. Gou¨ ezel,Martin boundary of random walks with unbounded jumps in hyperbolic groups, Ann. Probab. 43(2015), no. 5, 2374–2404

    2015

  23. [31]

    Gromov,Hyperbolic groups, inEssays in Group Theory, Math

    M. Gromov,Hyperbolic groups, inEssays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263

    1987

  24. [32]

    He,On the convergence of circle packings to the Riemann map, Invent

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

    1996

  25. [33]

    He,Rigidity of infinite disk patterns, Ann

    Z.-X. He,Rigidity of infinite disk patterns, Ann. of Math. (2)149(1999), no. 1, 1–33. 39

    1999

  26. [34]

    He and O

    Z.-X. He and O. Schramm,Fixed points, Koebe uniformization and circle packings, Ann. of Math. (2)137 (1993), no. 2, 369–406

    1993

  27. [35]

    He and O

    Z.-X. He and O. Schramm,Hyperbolic and parabolic packings, Discrete Comput. Geom.14(1995), no. 2, 123–149

    1995

  28. [36]

    Hutchcroft and Y

    T. Hutchcroft and Y. Peres,Boundaries of planar graphs: a unified approach, Electron. J. Probab.22 (2017), paper no. 100, 1–20

    2017

  29. [37]

    V. A. Kaimanovich and A. M. Vershik,Random walks on discrete groups: boundary and entropy, Ann. Probab.11(1983), no. 3, 457–490

    1983

  30. [38]

    V. A. Kaimanovich,The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2)152 (2000), no. 3, 659–692

    2000

  31. [39]

    Koebe,Kontaktprobleme der konformen Abbildung, Ber

    P. Koebe,Kontaktprobleme der konformen Abbildung, Ber. S¨ achs. Akad. Wiss. Leipzig Math.-Phys. Kl.88 (1936), 141–164

    1936

  32. [40]

    Liu,The Character of Thurston’s Circle Packings with Obtuse Exterior Intersection Angles, Anal

    C. Liu,The Character of Thurston’s Circle Packings with Obtuse Exterior Intersection Angles, Anal. Theory Appl.41(2025), no. 1, 80–98

    2025

  33. [41]

    Lyons and Y

    R. Lyons and Y. Peres,Probability on Trees and Networks, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016

    2016

  34. [42]

    Northshield,Square tilings, harmonic functions and electrical networks, Proc

    S. Northshield,Square tilings, harmonic functions and electrical networks, Proc. Amer. Math. Soc.114 (1992), no. 4, 1145–1152

    1992

  35. [43]

    Rivin and C

    I. Rivin and C. D. Hodgson,A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math.111(1993), 77–111

    1993

  36. [44]

    Rivin,Euclidean structures on simplicial surfaces and hyperbolic volume, Ann

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

    1994

  37. [45]

    Rivin,A characterization of ideal polyhedra in hyperbolic 3-space, Ann

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

    1996

  38. [46]

    Rodin and D

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

    1987

  39. [47]

    R. K. W. Roeder, J. H. Hubbard, and W. D. Dunbar,Andreev’s theorem on hyperbolic polyhedra, Ann. Inst. Fourier (Grenoble)57(2007), no. 3, 825–882

    2007

  40. [48]

    Schlenker,Hyperideal circle patterns, Math

    J.-M. Schlenker,Hyperideal circle patterns, Math. Res. Lett.12(2005), no. 1, 85–102

    2005

  41. [49]

    Schramm,Square tilings with prescribed combinatorics, Israel J

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

    1993

  42. [50]

    P. M. Soardi,Potential Theory on Infinite Networks, Lecture Notes in Mathematics, vol. 1590, Springer, Berlin, 1994

    1994

  43. [51]

    B. A. Springborn,Variational principles for circle patterns, Ph.D. thesis, Technische Universit¨ at Berlin, 2003

    2003

  44. [52]

    Stephenson,Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, Cambridge, 2005

    K. Stephenson,Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, Cambridge, 2005

    2005

  45. [53]

    W. P. Thurston,The geometry and topology of three-manifolds, Princeton lecture notes, 1979

    1979

  46. [54]

    Woess,Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, vol

    W. Woess,Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. Author information Huabin Ge Renmin University of China E-mail:hbge@ruc.edu.cn Chuwen Wang Renmin University of China E-mail:chuwenwang@...

    2000

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.