Pith. sign in

REVIEW 3 major objections 5 minor 72 references

Local convergence of random planar graphs

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Uniform random labelled planar graphs converge locally to a new infinite random graph, the UIPG.

desk verdict A major advance with a largely sound proof; the one load-bearing numerical inequality needs a rigorous bound. read the letter →

arxiv 1908.04850 v1 pith:3T4OE4VM submitted 2019-08-13 math.PR math.CO

Benedikt Stufler This is my paper · ORCID
classification math.PRmath.CO MSC 05C8060C0505C1060J8005A15
keywords randomplanargraphslocalweakconvergencequenchedlimituniforminfinitegraphTuttedecompositioncondensationGibbspartitionsenrichedtrees
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 that a graph drawn uniformly from all connected planar graphs on $n$ labelled vertices has a well-defined asymptotic local shape: the neighbourhood of a uniformly random vertex converges, in the stronger quenched sense, to a single infinite random planar graph called the uniform infinite planar graph (UIPG). It establishes the analogous quenched limits for uniform 2-connected planar graphs and for 2-connected planar maps, and it identifies how the UIPG is assembled from the 2-connected limit by attaching independent Boltzmann-distributed connected pieces. The proof runs through the Tutte decomposition probabilistically, encoding each connectivity layer as a tree that exhibits a condensation phenomenon, and the classical asymptotic count of planar graphs falls out as a by-product of the same machinery. A sympathetic reader should care because the UIPG provides a canonical infinite object in which local statistics of finite random planar graphs, such as degree distributions and subgraph counts, can be read off directly.

What carries the argument

The engine of the argument is a fully recursive tree-like encoding of the Tutte decomposition. For graphs the paper introduces the species $K$ and $R$ of networks, related by $K \equiv yR(x,K)$ with $R = J\,\mathrm{SEQ}(I^*)$, and for maps the analogous barred species $\bar{K}$ and $\bar{R}$; each identity turns the connectivity layer into a simply generated tree decorated by smaller networks. Sampling such an enriched tree produces a Galton-Watson tree with subexponential offspring in the subcritical regime, and the condensation phenomenon then forces a unique giant component at each layer, with fluctuations of order $n^{2/3}$ governed by a $3/2$-stable density. Local limit theorems for the giant component size and for fringe subtrees, together with Gibbs-partition transfer results, let quenched convergence pass successively from maps down to $\bar{O}$-, $\bar{R}$-, $\bar{K}$-, and $V$-cores, then back up the graph-side chain to 2-connected and connected planar graphs.

What would settle it

Compute certified interval bounds for $\nu_C = \rho_B \, \partial^2 B/\partial x^2(\rho_B,1)$ using rigorous interval arithmetic on the singular expansion of the 2-connected planar graph generating series; a certified lower bound at least 1 would disprove the condensation premise and invalidate the proof of Theorem 1.1, while a certified upper bound below 1 would close the remaining numerical gap.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: if $P_n$ is the uniform connected simple planar graph on $n$ labelled vertices and $v_n$ is a uniformly selected vertex, then the regular conditional law $\mathcal{L}((P_n,v_n)\mid P_n)$ converges in probability to the law of a limiting infinite planar graph $\hat{P}$, the UIPG, in the local topology. This is quenched convergence, meaning the empirical distribution of rooted neighbourhoods inside a single large random graph approximates the limit, not merely the averaged law. The paper also proves quenched local limits for uniform 2-connected planar graphs (with limit $\hat{B}$, the UI2PG) and for non-separable planar maps (with limit $\hat{V}$, the UI2PM), and it shows that $\hat{P}$ is obtained from $\hat{B}$ by inserting i.i.d. Boltzmann-distributed vertex-marked connected planar graphs at non-root vertices and a doubly marked Boltzmann component at the root. Along the way the paper recovers the asymptotic formula $p_n \sim c_G \rho_C^{-n} n^{-7/2}$ for the number of planar graphs, without the analytic integration used in the original proof.

Load-bearing premise

The whole condensation route for planar graphs rests on the strict inequality $\nu_C < 1$, where $\nu_C$ is a constant built from the generating series of 2-connected planar graphs; the paper checks this with approximate constants from an earlier enumeration paper, without rigorous error bounds, so if the true value reached 1 the condensation step and the main convergence argument would fail.

Editorial extensions

If this is right

  • The UIPG exists as a quenched local limit, and the stationary-rooted version of the same convergence implies that $\hat{P}$ is almost surely recurrent.
  • For any fixed finite connected graph $H$, the number of subgraph occurrences satisfies $\mathrm{emb}(H,P_n)/n \to \mathbb{E}[\mathrm{emb}^{\bullet}(H^{\bullet},\hat{P})]$ in probability.
  • The asymptotic enumeration constant and exponent $p_n \sim c_G\rho_C^{-n}n^{-7/2}$ follow from the probabilistic condensation argument without a single analytic integration step.
  • The vertex-weighted versions of random 2-connected planar graphs and non-separable planar maps admit quenched local limits with explicitly described infinite limiting objects.
  • The root degree of the UIPG matches the known asymptotic degree distribution of uniform random planar graphs.

Reading between the lines

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

  • Inference: because the route only uses a subexponential $n^{-5/2}$ census tail and Tutte stability, the same enriched-tree-plus-condensation scheme should produce quenched local limits for other Tutte-stable graph classes with the same census profile, and comparing the resulting limits would test how universal the UIPG-type shape is.
  • Inference: the paper explicitly notes that the graph-side decomposition $K \equiv yR(x,K)$ is not isomorphism-preserving, so the unlabelled random planar graph is not covered; controlling automorphism bias in that substitution would show whether the same UIPG appears for unlabelled graphs.
  • Inference: the proof identifies $n^{2/3}$-scale, $3/2$-stable fluctuations for the sizes of the successive giant cores; a concrete test is to generate moderately large planar graphs and check whether the largest 2-connected block size, rescaled by $n^{2/3}$, matches the stable density $h$ used throughout the paper.
  • Inference: quenched convergence suggests one can estimate UIPG subgraph probabilities from a single large sample graph, but the paper gives no rates; obtaining explicit rates would require quantitative versions of the condensation and transfer lemmas.
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

3 major / 5 minor

Summary. The paper develops a probabilistic framework for the local convergence of random planar structures. Its main theorem, Theorem 1.1, states that the uniform connected simple planar graph P_n on n labelled vertices, rooted at a uniformly chosen vertex, converges in the quenched sense in the local topology to a novel infinite random graph called the uniform infinite planar graph (UIPG). Along the way the paper establishes analogous quenched local limits for vertex-weighted 2-connected planar graphs (Theorem 9.11 and Theorem 1.2) and non-separable planar maps (Theorems 9.9 and 1.3), and it recovers the Giménez–Noy asymptotic formula for the number of planar graphs (Theorem 1.4). The proof combines Tutte's decomposition with Gibbs partitions, enriched tree encodings, condensation phenomena in simply generated trees, and transfer arguments between random mixtures.

Significance. If the technical gaps described below are closed, this would be a major contribution: it provides the first quenched local limit for uniform connected planar graphs, strengthening the existing annealed picture and yielding natural applications such as subgraph-count asymptotics via Corollary 1.5. The high-level architecture is coherent, and the paper contains a genuinely new probabilistic view of the Tutte decomposition, reducing planar graph limits to condensation in subcritical Galton–Watson trees and Gibbs partition convergence. The dependence on the author's previously published theorems is legitimate because those results are stated with explicit assumptions and are not fitted to the present problem. The main risk is not circularity but rather the completeness of several load-bearing technical estimates.

major comments (3)
  1. [Section 8.1, Eq. (8.5)] The inequality ν_C < 1 is load-bearing for Theorem 1.1, since E[ξ_P] = ν_C is the only source of subcriticality in Eq. (9.103), and Lemmas 3.2–3.3 are applied to the simply generated tree T^P_n only in that regime. The verification is currently an unchecked numerical evaluation: the singular expansion N(x,1) = D0 + D2X^2 + D3X^3 + O(X^4) is quoted from Bender et al. (2002) with approximate constants D0 ≈ 1.09417 and D2 ≈ −0.13749, and no explicit error bounds or validated enclosures are given for these constants or for ρ_B. A failure of ν_C < 1 would collapse the condensation argument, so the manuscript must either prove this inequality rigorously (for example, by interval arithmetic applied to the analytic expressions) or cite a verified computation with explicit error bounds; the displayed value 0.041302 < 1 is not itself a proof.
  2. [Section 9.1, Eq. (9.2) and following paragraph] The offspring distribution ξ_M of the simply generated tree encoding non-separable maps is supported on even integers, so it does not satisfy Condition (3.4) as stated. The manuscript asserts in one sentence that Lemmas 3.2 and 3.3 'may be extended' to this setting by rescaling by 1/2. These lemmas are used repeatedly, for example in Eqs. (9.3), (9.26), (9.52), and (9.75), to obtain the local limit theorems for core sizes, so the periodic extension is not a purely cosmetic detail. The paper should state and prove the even-version lemma, or explicitly spell out the rescaling argument and verify that the uniformity of the o(1) error terms is preserved.
  3. [Lemma 9.10, proof of Eq. (9.75)] The local limit theorem for O(K^n_t), which is later used in Sections 9.7–9.8 to derive quenched convergence of 2-connected planar graphs, depends on a long double-sum simplification. The proof delegates the main part to 'tedious but not difficult steps' and leaves the details to the reader. Because this estimate is the bridge from the local limit theorem for R(K^n_t) to that for O(K^n_t), the details should be included, or the reduction should be replaced by a direct derivation. As written, this is an assertion rather than a completed proof at a point that is load-bearing for Theorem 9.11.
minor comments (5)
  1. [Section 1.1, Introduction] The phrase 'by performing analytic integration and man m la using analytic methods' appears to contain a garbled or corrupted word ('man m la'); it should be corrected to a readable sentence.
  2. [Section 9.2, proof of Corollary 9.4] The notation ¯D(Mt_n) appears without definition; the context suggests it should be D(Mt_n), the D-network corresponding to V(Mt_n).
  3. [Remark 9.12] Equation (9.105) is asserted to follow from Eq. (9.104) 'by identical arguments' to Corollary 9.4, but the proof is not given, and the remark only describes verbally the subtle distinction between B(P_n) and the largest 2-connected block. Since this statement is not used in the proof of the main theorem, it could be moved to a clearly marked sketch or proved in full.
  4. [Section 8.4, Eq. (8.35)] The simplification leading to νM(t) would benefit from at least one intermediate algebraic step; as written, the reader must reproduce a lengthy reduction involving Eqs. (8.14), (8.15), (8.22), and (8.33).
  5. [Abstract and Theorem 1.4] The rendering 'ρ−n C' should be ρ_C^{-n} to match the notation in Eq. (1.4).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained given independent prior results and external enumeration inputs.

full rationale

The paper's main theorem is not circular. The quenched local limit of random connected planar graphs is derived through a chain: quenched local convergence of weighted planar maps (Lemma 9.1, Stufler 2019b) is passed down to non-separable maps, then to R-bar/O-bar cores using Gibbs partition and subcritical branching results (Stufler 2016, 2018, 2019a). These prior works are parameter-free with stated assumptions that do not include the present planar-graph limit, so they are independent support rather than self-referential inputs. The enumeration section (Section 8) uses the external asymptotic enumeration of 2-connected planar graphs by Bender et al. (2002) to verify the subcriticality condition nu_C < 1 via Eq. (8.5); the constants are approximate but they are not fitted in this paper, and the main probabilistic convergence does not reduce to the target formula. The recovery of the Gimenez-Noy formula is a derivation from Bender et al.'s 2-connected enumeration, not an input. The only notable caveat is the numerical, non-enclosure-based check of nu_C < 1 in Section 8.1, which is a rigor issue rather than circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

The paper introduces no fitted free parameters. Its central claims rest on standard Tutte/Whitney theory, on external analytic enumeration results (Bender et al.; Giménez et al.), on the author's own prior published theorems, and on two paper-specific assertions: the approximate verification of nu_C < 1 and the periodic extension of condensation lemmas. The latter two are not fully justified and are flagged as assumptions.

assumptions (8)
  • domain assumption Tutte's decomposition theory for simple graphs and planar maps
    Used in Sections 6 and 7 to derive the network decompositions N = S+P+H and D = S-bar+P-bar+H-bar; accepted from Cunningham-Edmonds, Hopcroft-Tarjan, Tutte, and Gagarin et al.
  • standard math Whitney's theorem on unique embeddings of 3-connected planar graphs
    Used to identify F0,1 = 1/2 F-bar0,1 and O = 1/2 O-bar in Equations (8.1) and (9.73).
  • domain assumption Bender, Gao and Wormald (2002) asymptotic enumeration of 2-connected planar graphs
    Provides [x^n]B(x,1) ~ c_B rho_B^{-n} n^{-7/2} and the singular expansion of N(x,1) used to verify nu_C < 1 and the enumeration transfer in Section 8.
  • domain assumption Giménez, Noy and Rué (2013, Lem. 6.6) mixture representation B(P_n) ≈ B^{rho_B}_{E_n}
    Used in Section 9.9 (Equation 9.106) to transfer quenched convergence from weighted 2-connected graphs to the core of P_n.
  • domain assumption Author's prior results: Stufler (2016, 2018, 2019a, 2019b)
    These provide Gibbs partition convergence, subcritical branching local limits, the quenched local limit for random planar maps M_t_n, and block-weighted graph transfer. They are published preprints with stated assumptions; treated as independent support.
  • ad hoc to paper Numerical inequality nu_C < 1 (Equation 8.5)
    The strict inequality is asserted from an approximate evaluation (0.041302 < 1) using D0, D2 from Bender et al. without rigorous error bounds. This is load-bearing for the condensation regime of the tree encoding of planar graphs.
  • ad hoc to paper Periodic extension of condensation lemmas to even offspring distributions
    Section 9.1 states that Lemma 3.2 and Lemma 3.3 hold for xi_M with only a rescaling argument; no proof is included.
  • standard math Subexponential density and big-jump random walk results (Denisov et al. 2008; Foss et al. 2013)
    Used in Sections 3 and 5 to justify coefficient asymptotics and Gibbs partition convergence.
invented entities (2)
  • Communities (H,A)
    purpose: Convergence-determining family of events used in Lemma 9.6 to carry out the induction when edge counts in the core neighbourhood are not preserved.
    Internal proof device; no independent empirical handle.
  • Semi-networks
    purpose: Describe how neighbourhoods of a root in K(M_t_n) intersect networks inserted at core edges.
    Internal proof device; variations on planar networks with relaxed connectivity requirement.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Local convergence of random planar graphs." pith.science (2026). https://pith.science/paper/3T4OE4VM

@misc{pith2026190804850,
  author       = {Pith},
  title        = {Pith review of: Local convergence of random planar graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3T4OE4VM}},
  note         = {Machine review of arXiv:1908.04850}
}
read the original abstract

The present work describes the asymptotic local shape of a graph drawn uniformly at random from all connected simple planar graphs with n labelled vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched limit in the local topology as n tends to infinity. We also establish such limits for random 2-connected planar graphs and maps as their number of edges tends to infinity. Our approach encompasses a new probabilistic view on the Tutte decomposition. This allows us to follow the path along the decomposition of connectivity from planar maps to planar graphs in a uniformed way, basing each step on condensation phenomena for random walks under subexponentiality and Gibbs partitions. Using large deviation results, we recover the asymptotic formula by Gim\'enez and Noy (2009) for the number of planar graphs.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

72 extracted references · 52 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archive author booktitle chapter doi edition editor eid eprint howpublished institution isbn issn journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'b...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    Local limits of conditioned G alton- W atson trees: the infinite spine case

    Romain Abraham and Jean-Fran c ois Delmas. Local limits of conditioned G alton- W atson trees: the infinite spine case. Electron. J. Probab. 19, no. 2, 19 (2014). ISSN 1083-6489

    work page 2014

  4. [4]

    Growing random 3-connected maps

    Louigi Addario-Berry. Growing random 3-connected maps. Electron. Commun. Probab. 19, no. 54, 12 (2014). ISSN 1083-589X. doi:10.1214/ECP.v19-3314. ://doi.org/10.1214/ECP.v19-3314

    work page doi:10.1214/ecp.v19-3314 2014

  5. [5]

    A probabilistic approach to block sizes in random maps

    Louigi Addario-Berry . A probabilistic approach to block sizes in random maps . ALEA Lat. Am. J. Probab. Math. Stat. 16 (1), 1--13 (2019)

    work page 2019

  6. [6]

    Uniform infinite planar triangulations

    Omer Angel and Oded Schramm. Uniform infinite planar triangulations. Comm. Math. Phys. 241 (2-3), 191--213 (2003). ISSN 0010-3616. doi:10.1007/978-1-4419-9675-6_16. ://dx.doi.org/10.1007/978-1-4419-9675-6_16

    doi:10.1007/978-1-4419-9675-6_16 2003

  7. [7]

    Random maps, coalescing saddles, singularity analysis, and A iry phenomena

    Cyril Banderier, Philippe Flajolet, Gilles Schaeffer and Mich \`e le Soria. Random maps, coalescing saddles, singularity analysis, and A iry phenomena. Random Structures Algorithms 19 (3-4), 194--246 (2001). ISSN 1042-9832. doi:10.1002/rsa.10021. ://dx.doi.org/10.1002/rsa.10021. Analysis of algorithms (Krynica Morska, 2000)

    work page doi:10.1002/rsa.10021 2001

  8. [8]

    Bender, Zhicheng Gao and Nicholas C

    Edward A. Bender, Zhicheng Gao and Nicholas C. Wormald. The number of labeled 2-connected planar graphs. Electron. J. Combin. 9 (1), Research Paper 43, 13 (2002). ISSN 1077-8926. ://www.combinatorics.org/Volume_9/Abstracts/v9i1r43.html

    work page 2002

Show all 72 references
  1. [9]

    Combinatorial species and tree-like structures, volume 67 of Encyclopedia of Mathematics and its Applications

    Fran c ois Bergeron, Gilbert Labelle and Pierre Leroux. Combinatorial species and tree-like structures, volume 67 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1998). ISBN 0-521-57323-8. Translated from the 1994 French original by ...

    1998

  2. [10]

    Convergence of probability measures

    Patrick Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition (1999). ISBN 0-471-19745-9. doi:10.1002/9780470316962. ://doi.org/10.1002/9780470316962. A Wiley-...

    1999 doi

  3. [11]

    o rnberg and Sigurdur \

    Jakob E. Bj \"o rnberg and Sigurdur \"O . Stef \'a nsson. Recurrence of bipartite planar maps. Electron. J. Probab. 19, no. 31, 40 (2014). ISSN 1083-6489. doi:10.1214/EJP.v19-3102. ://dx.doi.org/10.1214/EJP.v19-3102

    2014 doi

  4. [12]

    Planar maps as labeled mobiles

    J \'e r \'e mie Bouttier, Philippe Di Francesco and Emmanuel Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (1), Research Paper 69, 27 (2004). ISSN 1077-8926. ://www.combinatorics.org/Volume_11/Abstracts/v11i1r69.html

    2004

  5. [13]

    A note on conditional versus joint unconditional weak convergence in bootstrap consistency results

    Axel B \"u cher and Ivan Kojadinovic. A note on conditional versus joint unconditional weak convergence in bootstrap consistency results. Journal of Theoretical Probability (2018). ISSN 1572-9230. doi:10.1007/s10959-018-0823-3. ://doi.org/10.1007/s10959-018-0823-3

    2018 doi

  6. [14]

    On the diameter of random planar graphs

    Guillaume Chapuy, \' E ric Fusy, Omer Gim\' e nez and Marc Noy. On the diameter of random planar graphs. Combin. Probab. Comput. 24 (1), 145--178 (2015). ISSN 0963-5483. doi:10.1017/S0963548314000467. ://doi.org/10.1017/S0963548314000467

    2015 doi

  7. [15]

    A complete grammar for decomposing a family of graphs into 3-connected components

    Guillaume Chapuy, \' E ric Fusy, Mihyun Kang and Bilyana Shoilekova. A complete grammar for decomposing a family of graphs into 3-connected components. Electron. J. Combin. 15 (1), Research Paper 148, 39 (2008). ISSN 1077-8926. ://www.combinatorics.org/Volume_15/Abstracts/v15i...

    2008

  8. [16]

    Functions of probability measures

    Joshua Chover, Peter Ney and Stephen Wainger. Functions of probability measures. J. Analyse Math. 26, 255--302 (1973). ISSN 0021-7670

    1973

  9. [17]

    Cunningham and Jack Edmonds

    William H. Cunningham and Jack Edmonds. A combinatorial decomposition theory. Canad. J. Math. 32 (3), 734--765 (1980). ISSN 0008-414X. doi:10.4153/CJM-1980-057-7. ://doi.org/10.4153/CJM-1980-057-7

    1980 doi

  10. [18]

    Random graphs - the local convergence point of view (2018)

    Nicolas Curien. Random graphs - the local convergence point of view (2018). ://www.math.u-psud.fr/ curien/cours/cours-RG.pdf

    2018

  11. [19]

    A view from infinity of the uniform infinite planar quadrangulation

    Nicolas Curien, Laurent M \'e nard and Gr \'e gory Miermont. A view from infinity of the uniform infinite planar quadrangulation. ALEA Lat. Am. J. Probab. Math. Stat. 10 (1), 45--88 (2013). ISSN 1980-0436

    2013

  12. [20]

    Alain Denise, Marcio Vasconcellos and Dominic J. A. Welsh. The random planar graph. Congr. Numer. 113, 61--79 (1996). ISSN 0384-9864. Festschrift for C. St. J. A. Nash-Williams

    1996

  13. [21]

    Dieker and Vsevolod Shneer

    Denis Denisov, Antonius B. Dieker and Vsevolod Shneer. Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36 (5), 1946--1991 (2008). ISSN 0091-1798. doi:10.1214/07-AOP382. ://dx.doi.org/10.1214/07-AOP382

    2008 doi

  14. [22]

    Random trees

    Michael Drmota. Random trees. SpringerWienNewYork, Vienna (2009). ISBN 978-3-211-75355-2. doi:10.1007/978-3-211-75357-6. ://dx.doi.org/10.1007/978-3-211-75357-6. An interplay between combinatorics and probability

    2009 doi

  15. [23]

    Degree distribution in random planar graphs

    Michael Drmota, Omer Gim\' e nez and Marc Noy. Degree distribution in random planar graphs. J. Combin. Theory Ser. A 118 (7), 2102--2130 (2011). ISSN 0097-3165. doi:10.1016/j.jcta.2011.04.010. ://doi.org/10.1016/j.jcta.2011.04.010

    2011 doi

  16. [24]

    Michael Drmota, Omer Gim\' e nez, Marc Noy, Konstantinos Panagiotou and A. Steger. The maximum degree of random planar graphs. Proc. Lond. Math. Soc. (3) 109 (4), 892--920 (2014). ISSN 0024-6115. doi:10.1112/plms/pdu024. ://doi.org/10.1112/plms/pdu024

    2014 doi

  17. [25]

    A central limit theorem for the number of degree- k vertices in random maps

    Michael Drmota and Konstantinos Panagiotou. A central limit theorem for the number of degree- k vertices in random maps. Algorithmica 66 (4), 741--761 (2013). ISSN 0178-4617. ://doi.org/10.1007/s00453-013-9751-x

    2013 doi

  18. [26]

    Pattern occurrences in random planar maps

    Michael Drmota and Benedikt Stufler . Pattern occurrences in random planar maps . arXiv e-prints arXiv:1801.10007 (2018). 1801.10007

    2018 arXiv

  19. [27]

    The number of double triangles in random planar maps

    Michael Drmota and Guan-Ru Yu. The number of double triangles in random planar maps. Proceedings AofA 2018. Leibniz International Proceedings in Informatics. 110, 19:1--19:18 (2018)

    2018

  20. [28]

    Functions of power series

    Paul Embrechts and Edward Omey. Functions of power series. Yokohama Math. J. 32 (1-2), 77--88 (1984). ISSN 0044-0523

    1984

  21. [29]

    Analytic combinatorics

    Philippe Flajolet and Robert Sedgewick. Analytic combinatorics. Cambridge University Press, Cambridge (2009). ISBN 978-0-521-89806-5. doi:10.1017/CBO9780511801655. ://dx.doi.org/10.1017/CBO9780511801655

    2009 doi

  22. [30]

    An introduction to heavy-tailed and subexponential distributions

    Sergey Foss, Dmitry Korshunov and Stan Zachary. An introduction to heavy-tailed and subexponential distributions. Springer Series in Operations Research and Financial Engineering. Springer, New York, second edition (2013). ISBN 978-1-4614-7100-4; 978-1-4614-7101-1. doi:10.1007...

    2013 doi

  23. [31]

    Structure and enumeration of two-connected graphs with prescribed three-connected components

    Andrei Gagarin, Gilbert Labelle, Pierre Leroux and Timothy Walsh. Structure and enumeration of two-connected graphs with prescribed three-connected components. Adv. in Appl. Math. 43 (1), 46--74 (2009). ISSN 0196-8858. doi:10.1016/j.aam.2009.01.002. ://doi.org/10.1016/j.aam.20...

    2009 doi

  24. [32]

    On the number of edges in random planar graphs

    Stefanie Gerke and Colin McDiarmid. On the number of edges in random planar graphs. Combin. Probab. Comput. 13 (2), 165--183 (2004). ISSN 0963-5483. doi:10.1017/S0963548303005947. ://doi.org/10.1017/S0963548303005947

    2004 doi

  25. [33]

    Asymptotic enumeration and limit laws of planar graphs

    Omer Gim\' e nez and Marc Noy. Asymptotic enumeration and limit laws of planar graphs. J. Amer. Math. Soc. 22 (2), 309--329 (2009). ISSN 0894-0347. doi:10.1090/S0894-0347-08-00624-3. ://doi.org/10.1090/S0894-0347-08-00624-3

    2009 doi

  26. [34]

    Graph classes with given 3-connected components: asymptotic enumeration and random graphs

    Omer Gim\' e nez, Marc Noy and Juanjo Ru\' e . Graph classes with given 3-connected components: asymptotic enumeration and random graphs. Random Structures Algorithms 42 (4), 438--479 (2013). ISSN 1042-9832. doi:10.1002/rsa.20421. ://doi.org/10.1002/rsa.20421

    2013 doi

  27. [35]

    Recurrence of planar graph limits

    Ori Gurel-Gurevich and Asaf Nachmias. Recurrence of planar graph limits. Ann. of Math. (2) 177 (2), 761--781 (2013). ISSN 0003-486X. doi:10.4007/annals.2013.177.2.10. ://dx.doi.org/10.4007/annals.2013.177.2.10

    2013 doi

  28. [36]

    Frank Harary and Edgar M. Palmer. Graphical enumeration. Academic Press, New York-London (1973)

    1973

  29. [37]

    J. E. Hopcroft and R. E. Tarjan. Dividing a graph into triconnected components. SIAM J. Comput. 2, 135--158 (1973). ISSN 0097-5397. doi:10.1137/0202012. ://doi.org/10.1137/0202012

    1973 doi

  30. [38]

    Simply generated trees, conditioned G alton- W atson trees, random allocations and condensation

    Svante Janson. Simply generated trees, conditioned G alton- W atson trees, random allocations and condensation. Probab. Surv. 9, 103--252 (2012). ISSN 1549-5787. doi:10.1214/11-PS188. ://dx.doi.org/10.1214/11-PS188

    2012 doi

  31. [39]

    Condensation in nongeneric trees

    Thordur Jonsson and Sigurdur \"O rn Stef \'a nsson. Condensation in nongeneric trees. J. Stat. Phys. 142 (2), 277--313 (2011). ISSN 0022-4715. doi:10.1007/s10955-010-0104-8. ://dx.doi.org/10.1007/s10955-010-0104-8

    2011 doi

  32. [40]

    Une th\'eorie combinatoire des s\'eries formelles

    Andr \'e Joyal. Une th\'eorie combinatoire des s\'eries formelles. Adv. in Math. 42 (1), 1--82 (1981). ISSN 0001-8708. doi:10.1016/0001-8708(81)90052-9. ://dx.doi.org/10.1016/0001-8708(81)90052-9

    1981 doi

  33. [41]

    Limit theorems for conditioned non-generic G alton- W atson trees

    Igor Kortchemski. Limit theorems for conditioned non-generic G alton- W atson trees. Ann. Inst. Henri Poincar\'e Probab. Stat. 51 (2), 489--511 (2015). ISSN 0246-0203. doi:10.1214/13-AIHP580. ://dx.doi.org/10.1214/13-AIHP580

    2015 doi

  34. [42]

    Local structure of random quadrangulations

    Maxim Krikun . Local structure of random quadrangulations . ArXiv Mathematics e-prints (2005). math/0512304

    2005 arXiv

  35. [43]

    On local weak limit and subgraph counts for sparse random graphs

    Valentas Kurauskas . On local weak limit and subgraph counts for sparse random graphs . ArXiv e-prints (2015). 1504.08103

    2015 arXiv

  36. [44]

    Une nouvelle d\'emonstration combinatoire des formules d'inversion de L agrange

    Gilbert Labelle. Une nouvelle d\'emonstration combinatoire des formules d'inversion de L agrange. Adv. in Math. 42 (3), 217--247 (1981). ISSN 0001-8708. doi:10.1016/0001-8708(81)90041-4. ://dx.doi.org/10.1016/0001-8708(81)90041-4

    1981 doi

  37. [45]

    Liskovets

    Valery A. Liskovets. A pattern of asymptotic vertex valency distributions in planar maps. J. Combin. Theory Ser. B 75 (1), 116--133 (1999). ISSN 0095-8956. doi:10.1006/jctb.1998.1870. ://doi.org/10.1006/jctb.1998.1870

    1999

  38. [46]

    A structural characterization of planar combinatorial graphs

    Saunders Mac Lane. A structural characterization of planar combinatorial graphs. Duke Math. J. 3 (3), 460--472 (1937). ISSN 0012-7094. doi:10.1215/S0012-7094-37-00336-3. ://doi.org/10.1215/S0012-7094-37-00336-3

    1937 doi

  39. [47]

    Random graphs on surfaces

    Colin McDiarmid. Random graphs on surfaces. J. Combin. Theory Ser. B 98 (4), 778--797 (2008). ISSN 0095-8956. doi:10.1016/j.jctb.2007.11.006. ://dx.doi.org/10.1016/j.jctb.2007.11.006

    2008 doi

  40. [48]

    Random graphs from a minor-closed class

    Colin McDiarmid. Random graphs from a minor-closed class. Combin. Probab. Comput. 18 (4), 583--599 (2009). ISSN 0963-5483. doi:10.1017/S0963548309009717. ://dx.doi.org/10.1017/S0963548309009717

    2009 doi

  41. [49]

    Random graphs from a weighted minor-closed class

    Colin McDiarmid. Random graphs from a weighted minor-closed class. Electron. J. Combin. 20 (2), Paper 52, 39 (2013). ISSN 1077-8926

    2013

  42. [50]

    Colin McDiarmid, Angelika Steger and Dominic J. A. Welsh. Random planar graphs. J. Combin. Theory Ser. B 93 (2), 187--205 (2005). ISSN 0095-8956. doi:10.1016/j.jctb.2004.09.007. ://dx.doi.org/10.1016/j.jctb.2004.09.007

    2005 doi

  43. [51]

    Percolation on uniform infinite planar maps

    Laurent M \'e nard and Pierre Nolin. Percolation on uniform infinite planar maps. Electron. J. Probab. 19, no. 79, 27 (2014). ISSN 1083-6489. doi:10.1214/EJP.v19-2675. ://dx.doi.org/10.1214/EJP.v19-2675

    2014 doi

  44. [52]

    Graphs on surfaces

    Bojan Mohar and Carsten Thomassen. Graphs on surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD (2001). ISBN 0-8018-6689-8

    2001

  45. [53]

    Mullin and Paul

    Ronald C. Mullin and Paul. J. Schellenberg. The enumeration of c -nets via quadrangulations. J. Combinatorial Theory 4, 259--276 (1968)

    1968

  46. [54]

    Random planar graphs and beyond

    Marc Noy. Random planar graphs and beyond. Proc. ICM (2014)

    2014

  47. [55]

    Further results on random cubic planar graphs

    Marc Noy , Cl \'e ment Requil \'e and Juanjo Ru \'e . Further results on random cubic planar graphs . arXiv e-prints arXiv:1802.06679 (2018). 1802.06679

    2018 arXiv

  48. [56]

    u rgen Pr\

    Deryk Osthus, Hans J\" u rgen Pr\" o mel and Anusch Taraz. On random planar graphs, the number of planar graphs and their triangulations. J. Combin. Theory Ser. B 88 (1), 119--134 (2003). ISSN 0095-8956. doi:10.1016/S0095-8956(02)00040-0. ://doi.org/10.1016/S0095-8956(02)00040-0

    2003 doi

  49. [57]

    On the degree distribution of random planar graphs

    Konstantinos Panagiotou and Angelika Steger. On the degree distribution of random planar graphs. In Proceedings of the T wenty- S econd A nnual ACM - SIAM S ymposium on D iscrete A lgorithms , pages 1198--1210. SIAM, Philadelphia, PA (2011)

    2011

  50. [58]

    Scaling limits of random graphs from subcritical classes

    Konstantinos Panagiotou, Benedikt Stufler and Kerstin Weller. Scaling limits of random graphs from subcritical classes. Ann. Probab. 44 (5), 3291--3334 (2016). ISSN 0091-1798. doi:10.1214/15-AOP1048

    2016 doi

  51. [59]

    Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics

    Jim Pitman. Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2006). ISBN 978-3-540-30990-1; 3-540-30990-X. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7--24, 2002, With a foreword...

    2006

  52. [60]

    A course on large deviations with an introduction to G ibbs measures , volume 162 of Graduate Studies in Mathematics

    Firas Rassoul-Agha and Timo Sepp\" a l\" a inen. A course on large deviations with an introduction to G ibbs measures , volume 162 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2015). ISBN 978-0-8218-7578-0

    2015

  53. [61]

    Local convergence of large critical multi-type G alton- W atson trees and applications to random maps

    Robin Stephenson. Local convergence of large critical multi-type G alton- W atson trees and applications to random maps. J. Theoret. Probab. 31 (1), 159--205 (2018). ISSN 0894-9840. doi:10.1007/s10959-016-0707-3. ://doi.org/10.1007/s10959-016-0707-3

    2018 doi

  54. [62]

    Limits of random tree-like discrete structures

    Benedikt Stufler . Limits of random tree-like discrete structures . ArXiv e-prints (2016). 1612.02580

    2016 arXiv

  55. [63]

    Gibbs partitions: The convergent case

    Benedikt Stufler. Gibbs partitions: The convergent case. Random Structures & Algorithms 53 (3), 537--558 (2018). doi:10.1002/rsa.20771. ://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.20771. https://onlinelibrary.wiley.com/doi/pdf/10.1002/rsa.20771

    2018 doi

  56. [64]

    Local limits of large G alton-- W atson trees rerooted at a random vertex

    Benedikt Stufler. Local limits of large G alton-- W atson trees rerooted at a random vertex. Ann. Inst. H. Poincaré Probab. Statist. 55 (1), 155--183 (2019). doi:10.1214/17-AIHP879. ://doi.org/10.1214/17-AIHP879

    2019 doi

  57. [65]

    On the maximal offspring in a subcritical branching process

    Benedikt Stufler . On the maximal offspring in a subcritical branching process . arXiv e-prints arXiv:1901.04603 (2019 a ). 1901.04603

    1901 arXiv

  58. [66]

    Rerooting multi-type branching trees: the infinite spine case

    Benedikt Stufler . Rerooting multi-type branching trees: the infinite spine case . arXiv e-prints (2019 b )

    2019

  59. [67]

    A theory of 3 -connected graphs

    William Thomas Tutte. A theory of 3 -connected graphs. Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23, 441--455 (1961)

    1961

  60. [68]

    A census of planar maps

    William Thomas Tutte. A census of planar maps. Canad. J. Math. 15, 249--271 (1963). ISSN 0008-414X

    1963

  61. [69]

    Connectivity in graphs

    William Thomas Tutte. Connectivity in graphs. Mathematical Expositions, No. 15. University of Toronto Press, Toronto, Ont.; Oxford University Press, London (1966)

    1966

  62. [70]

    Graph theory, volume 21 of Encyclopedia of Mathematics and its Applications

    William Thomas Tutte. Graph theory, volume 21 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA (1984). ISBN 0-201-13520-5. With a foreword by C. St. J. A. Nash-Williams

    1984

  63. [71]

    2- I somorphic G raphs

    Hassler Whitney. 2- I somorphic G raphs. Amer. J. Math. 55 (1-4), 245--254 (1933). ISSN 0002-9327. doi:10.2307/2371127. ://doi.org/10.2307/2371127

    1933 doi

  64. [72]

    @lbibitem[#1]#2 [\@biblabel #1 ] @tempwidthb \@biblabel #1 @tempwidthb> @tempwidtha @tempwidtha= @tempwidthb @filesw \@auxout #2 #1 @bibitem#1 @filesw \@auxout #1 \@listctr @tempwidthb \@biblabel @tempwidthb> @tempwidtha @tempwidtha= @tempwidthb \@lbibitem @lbibitem \@bibitem ...

    2000

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.