Open Access
2024 A phase transition in block-weighted random maps
William Fleurat, Zéphyr Salvy
Author Affiliations +
Electron. J. Probab. 29: 1-61 (2024). DOI: 10.1214/24-EJP1089
Abstract

We consider the model of random planar maps of size n biased by a weight u>0 per 2-connected block, and the closely related model of random planar quadrangulations of size n biased by a weight u>0 per simple component. We exhibit a phase transition at the critical value uC=95. If u<uC, a condensation phenomenon occurs: the largest block is of size Θ(n). Moreover, for quadrangulations we show that the diameter is of order n14, and the scaling limit is the Brownian sphere. When u>uC, the largest block is of size Θ(log(n)), the scaling order for distances is n12, and the scaling limit is the Brownian tree. Finally, for u=uC, the largest block is of size Θ(n23), the scaling order for distances is n13, and the scaling limit is the stable tree of parameter 32.

References

1.

Louigi Addario-Berry, A probabilistic approach to block sizes in random maps, ALEA – Latin American Journal of Probability and Mathematical Statistics XVI (2019), 1–13.  MR3903022Louigi Addario-Berry, A probabilistic approach to block sizes in random maps, ALEA – Latin American Journal of Probability and Mathematical Statistics XVI (2019), 1–13.  MR3903022

2.

Louigi Addario-Berry and Marie Albenque, The scaling limit of random simple triangulations and random simple quadrangulations, The Annals of Probability 45 (2017), no. 5, 2767–2825.  MR3706731Louigi Addario-Berry and Marie Albenque, The scaling limit of random simple triangulations and random simple quadrangulations, The Annals of Probability 45 (2017), no. 5, 2767–2825.  MR3706731

3.

Louigi Addario-Berry and Marie Albenque, Convergence of odd-angulations via symmetrization of labeled trees, Annales Henri Lebesgue 4 (2021), 653–683.  MR4315765Louigi Addario-Berry and Marie Albenque, Convergence of odd-angulations via symmetrization of labeled trees, Annales Henri Lebesgue 4 (2021), 653–683.  MR4315765

4.

Louigi Addario-Berry and Yuting Wen, Joint convergence of random quadrangulations and their cores, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53 (2017), no. 4, 1890–1920.  MR3729639Louigi Addario-Berry and Yuting Wen, Joint convergence of random quadrangulations and their cores, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53 (2017), no. 4, 1890–1920.  MR3729639

5.

Marie Albenque and Jean-Francois Marckert, Some families of increasing planar maps., Electronic Journal of Probability 13 (2008), 1624–1671 (eng).  MR2438817Marie Albenque and Jean-Francois Marckert, Some families of increasing planar maps., Electronic Journal of Probability 13 (2008), 1624–1671 (eng).  MR2438817

6.

David Aldous, The Continuum Random Tree III, The Annals of Probability 21 (1993), no. 1, 248–289.  MR1207226David Aldous, The Continuum Random Tree III, The Annals of Probability 21 (1993), no. 1, 248–289.  MR1207226

7.

Eleanor Archer, Random walks on decorated galton-watson trees, Ph.D. thesis, University of Warwick, June 2020.Eleanor Archer, Random walks on decorated galton-watson trees, Ph.D. thesis, University of Warwick, June 2020.

8.

Inés Armendáriz and Michail Loulakis, Conditional distribution of heavy tailed random variables on large deviations of their sum, Stochastic Processes and their Applications 121 (2009), 1138–1147.  MR2775110Inés Armendáriz and Michail Loulakis, Conditional distribution of heavy tailed random variables on large deviations of their sum, Stochastic Processes and their Applications 121 (2009), 1138–1147.  MR2775110

9.

Cyril Banderier, Philippe Flajolet, Gilles Schaeffer, and Michèle Soria, Random Maps, Coalescing Saddles, Singularity Analysis, and Airy Phenomena, Random Struct. Algorithms 19 (2001), no. 3-4, 194–246.  MR1871555Cyril Banderier, Philippe Flajolet, Gilles Schaeffer, and Michèle Soria, Random Maps, Coalescing Saddles, Singularity Analysis, and Airy Phenomena, Random Struct. Algorithms 19 (2001), no. 3-4, 194–246.  MR1871555

10.

J. Bertoin, Lévy processes, Cambridge Tracts in Mathematics, Cambridge University Press, 1996.  MR1406564J. Bertoin, Lévy processes, Cambridge Tracts in Mathematics, Cambridge University Press, 1996.  MR1406564

11.

Jérémie Bettinelli, Scaling limit of random planar quadrangulations with a boundary, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 2, 432–477.  MR3335010Jérémie Bettinelli, Scaling limit of random planar quadrangulations with a boundary, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 2, 432–477.  MR3335010

12.

Jérémie Bettinelli, Emmanuel Jacob, and Grégory Miermont, The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection, Electronic Journal of Probability 19 (2014), 1–16.  MR3256874Jérémie Bettinelli, Emmanuel Jacob, and Grégory Miermont, The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection, Electronic Journal of Probability 19 (2014), 1–16.  MR3256874

13.

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, June 1989 (English).  MR1015093N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, June 1989 (English).  MR1015093

14.

Valentin Bonzom, Large N limits in tensor models: Towards more universality classes of colored triangulations in dimension d2, Symmetry, Integrability and Geometry: Methods and Applications 12 (2016), no. 073, 39.  MR3528381Valentin Bonzom, Large N limits in tensor models: Towards more universality classes of colored triangulations in dimension d2, Symmetry, Integrability and Geometry: Methods and Applications 12 (2016), no. 073, 39.  MR3528381

15.

J Bouttier and E Guitter, Distance statistics in quadrangulations with no multiple edges and the geometry of minbus, Journal of Physics A: Mathematical and Theoretical 43 (2010), no. 20, 205207.  MR2639918J Bouttier and E Guitter, Distance statistics in quadrangulations with no multiple edges and the geometry of minbus, Journal of Physics A: Mathematical and Theoretical 43 (2010), no. 20, 205207.  MR2639918

16.

William G. Brown, Enumeration of quadrangular dissections of the disk, Canadian Journal of Mathematics 17 (1965), 302–317.  MR0186571William G. Brown, Enumeration of quadrangular dissections of the disk, Canadian Journal of Mathematics 17 (1965), 302–317.  MR0186571

17.

Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.  MR1835418Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.  MR1835418

18.

Alessandra Caraceni, The scaling limit of random outerplanar maps, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 52 (2016), no. 4, 1667–1686.  MR3573291Alessandra Caraceni, The scaling limit of random outerplanar maps, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 52 (2016), no. 4, 1667–1686.  MR3573291

19.

Ariane Carrance, Convergence of Eulerian triangulations, Electronic Journal of Probability (2021), no. 26, 1–48.  MR4235469Ariane Carrance, Convergence of Eulerian triangulations, Electronic Journal of Probability (2021), no. 26, 1–48.  MR4235469

20.

Guillaume Chapuy, Éric Fusy, Omer Giménez, and Marc Noy, On the diameter of random planar graphs, Combinatorics, Probability and Computing 24 (2015), no. 1, 145–178.  MR3318042Guillaume Chapuy, Éric Fusy, Omer Giménez, and Marc Noy, On the diameter of random planar graphs, Combinatorics, Probability and Computing 24 (2015), no. 1, 145–178.  MR3318042

21.

Philippe Chassaing and Gilles Schaeffer, Random planar lattices and integrated superbrownian excursion, Probability Theory and Related Fields 128 (2004), no. 2, 161–212.  MR2031225Philippe Chassaing and Gilles Schaeffer, Random planar lattices and integrated superbrownian excursion, Probability Theory and Related Fields 128 (2004), no. 2, 161–212.  MR2031225

22.

Oliver Cooley, Wenjie Fang, Nicola Del Giudice, and Mihyun Kang, Subcritical random hypergraphs, high-order components, and hypertrees, SIAM Journal on Discrete Mathematics 34 (2020), no. 4, 2033–2062.  MR4159560Oliver Cooley, Wenjie Fang, Nicola Del Giudice, and Mihyun Kang, Subcritical random hypergraphs, high-order components, and hypertrees, SIAM Journal on Discrete Mathematics 34 (2020), no. 4, 2033–2062.  MR4159560

23.

Nicolas Curien and Igor Kortchemski, Percolation on random triangulations and stable looptrees, Probability Theory and Related Fields 163 (2013).  MR3405619Nicolas Curien and Igor Kortchemski, Percolation on random triangulations and stable looptrees, Probability Theory and Related Fields 163 (2013).  MR3405619

24.

Nicolas Curien and Igor Kortchemski, Random stable looptrees, Electronic Journal of Probability 19 (2013).  MR3286462Nicolas Curien and Igor Kortchemski, Random stable looptrees, Electronic Journal of Probability 19 (2013).  MR3286462

25.

R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002.  MR1932358R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002.  MR1932358

26.

Bertrand Duplantier and Scott Sheffield, Duality and the Knizhnik-Polyakov-Zamolodchikov relation in Liouville quantum gravity, Phys. Rev. Lett. 102 (2009), 150603.  MR2501276Bertrand Duplantier and Scott Sheffield, Duality and the Knizhnik-Polyakov-Zamolodchikov relation in Liouville quantum gravity, Phys. Rev. Lett. 102 (2009), 150603.  MR2501276

27.

Thomas Duquesne, A limit theorem for the contour process of conditioned Galton–Watson trees, The Annals of Probability 31 (2003), no. 2, 996–1027.  MR1964956Thomas Duquesne, A limit theorem for the contour process of conditioned Galton–Watson trees, The Annals of Probability 31 (2003), no. 2, 996–1027.  MR1964956

28.

William Feller, An introduction to probability theory and its applications, 2 ed., vol. 2, Wiley, 1971.  MR0386954William Feller, An introduction to probability theory and its applications, 2 ed., vol. 2, Wiley, 1971.  MR0386954

29.

Philippe Flajolet and Robert Sedgewick, Analytic combinatorics, Cambridge University Press, 2009.  MR2825582Philippe Flajolet and Robert Sedgewick, Analytic combinatorics, Cambridge University Press, 2009.  MR2825582

30.

Frank Harary, Graph theory, Addison-Wesley series in mathematics, Addison-Wesley Publishing Company, 1969.  MR0256911Frank Harary, Graph theory, Addison-Wesley series in mathematics, Addison-Wesley Publishing Company, 1969.  MR0256911

31.

I. A. Ibragimov, Independent and stationary sequences of random variables, Wolters-Noordhoff, Groningen, January 1971 (English).  MR0322926I. A. Ibragimov, Independent and stationary sequences of random variables, Wolters-Noordhoff, Groningen, January 1971 (English).  MR0322926

32.

Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Springer Berlin Heidelberg, Berlin, Heidelberg, 1987.  MR0959133Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Springer Berlin Heidelberg, Berlin, Heidelberg, 1987.  MR0959133

33.

Svante Janson, Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation, Probability Surveys 9 (2012), 103–252.  MR2908619Svante Janson, Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation, Probability Surveys 9 (2012), 103–252.  MR2908619

34.

Svante Janson and Sigurdur Örn Stefánsson, Scaling limits of random planar maps with a unique large face, The Annals of Probability 43 (2015), no. 3, 1045–1081.  MR3342658Svante Janson and Sigurdur Örn Stefánsson, Scaling limits of random planar maps with a unique large face, The Annals of Probability 43 (2015), no. 3, 1045–1081.  MR3342658

35.

Thordur Jonsson and Sigurdur Örn Stefánsson, Condensation in nongeneric trees, Journal of Statistical Physics 142 (2011), no. 2, 277–313.  MR2764126Thordur Jonsson and Sigurdur Örn Stefánsson, Condensation in nongeneric trees, Journal of Statistical Physics 142 (2011), no. 2, 277–313.  MR2764126

36.

Igor Kortchemski, A simple proof of Duquesne’s theorem on contour processes of conditioned Galton–Watson trees, pp. 537–558, Springer International Publishing, Heidelberg, 2013.  MR3185928Igor Kortchemski, A simple proof of Duquesne’s theorem on contour processes of conditioned Galton–Watson trees, pp. 537–558, Springer International Publishing, Heidelberg, 2013.  MR3185928

37.

Igor Kortchemski, Limit theorems for conditioned non-generic Galton–Watson trees, 51 (2015), no. 2, 489–511.  MR3335012Igor Kortchemski, Limit theorems for conditioned non-generic Galton–Watson trees, 51 (2015), no. 2, 489–511.  MR3335012

38.

Igor Kortchemski and Cyril Marzouk, Large deviation local limit theorems and limits of biconditioned trees and maps, 2021.  MR4663496Igor Kortchemski and Cyril Marzouk, Large deviation local limit theorems and limits of biconditioned trees and maps, 2021.  MR4663496

39.

Igor Kortchemski and Loïc Richier, The boundary of random planar maps via looptrees, Annales de la Faculté des sciences de Toulouse: Mathématiques Ser. 6, 29 (2020), no. 2, 391–430 (en).  MR4150547Igor Kortchemski and Loïc Richier, The boundary of random planar maps via looptrees, Annales de la Faculté des sciences de Toulouse: Mathématiques Ser. 6, 29 (2020), no. 2, 391–430 (en).  MR4150547

40.

Jean-François Le Gall, Random real trees, Annales de la Faculté des sciences de Toulouse: Mathématiques Ser. 6, 15 (2006), no. 1, 35–62 (en).  MR2225746Jean-François Le Gall, Random real trees, Annales de la Faculté des sciences de Toulouse: Mathématiques Ser. 6, 15 (2006), no. 1, 35–62 (en).  MR2225746

41.

Jean-François Le Gall, The topological structure of scaling limits of large planar maps, Inventiones mathematicae 169 (2007), no. 3, 621–670.  MR2336042Jean-François Le Gall, The topological structure of scaling limits of large planar maps, Inventiones mathematicae 169 (2007), no. 3, 621–670.  MR2336042

42.

Jean-François Le Gall, Geodesics in large planar maps and in the Brownian map, Acta Mathematica 205 (2010), no. 2, 287–360.  MR2746349Jean-François Le Gall, Geodesics in large planar maps and in the Brownian map, Acta Mathematica 205 (2010), no. 2, 287–360.  MR2746349

43.

Jean-François Le Gall, Uniqueness and universality of the Brownian map, The Annals of Probability 41 (2013), no. 4, 2880–2960.  MR3112934Jean-François Le Gall, Uniqueness and universality of the Brownian map, The Annals of Probability 41 (2013), no. 4, 2880–2960.  MR3112934

44.

Jean-François Le Gall and Grégory Miermont, Scaling limits of random planar maps with large faces, The Annals of Probability 39 (2011), no. 1, 1–69.  MR2778796Jean-François Le Gall and Grégory Miermont, Scaling limits of random planar maps with large faces, The Annals of Probability 39 (2011), no. 1, 1–69.  MR2778796

45.

Thomas Lehéricy, First-passage percolation in random planar maps and Tutte’s bijection, Electronic Journal of Probability (2022).  MR4387838Thomas Lehéricy, First-passage percolation in random planar maps and Tutte’s bijection, Electronic Journal of Probability (2022).  MR4387838

46.

Luca Lionni, Colored discrete spaces: Higher dimensional combinatorial maps and quantum gravity, Ph.D. thesis, Université Paris Saclay (COmUE), 2017.  MR3822340Luca Lionni, Colored discrete spaces: Higher dimensional combinatorial maps and quantum gravity, Ph.D. thesis, Université Paris Saclay (COmUE), 2017.  MR3822340

47.

Russell Lyons, Robin Pemantle, and Yuval Peres, Conceptual proofs of LlogL criteria for mean behavior of branching processes, Annals of Probability 23 (1995), 1125–1138.  MR1349164Russell Lyons, Robin Pemantle, and Yuval Peres, Conceptual proofs of LlogL criteria for mean behavior of branching processes, Annals of Probability 23 (1995), 1125–1138.  MR1349164

48.

Jean-François Marckert and Abdelkader Mokkadem, The depth first processes of Galton-Watson trees converge to the same Brownian excursion, The Annals of Probability 31 (2003), no. 3, 1655–1678.  MR1989446Jean-François Marckert and Abdelkader Mokkadem, The depth first processes of Galton-Watson trees converge to the same Brownian excursion, The Annals of Probability 31 (2003), no. 3, 1655–1678.  MR1989446

49.

Cyril Marzouk, Scaling limits of random bipartite planar maps with a prescribed degree sequence, Random Structures & Algorithms 53 (2018), no. 3, 448–503.  MR3854042Cyril Marzouk, Scaling limits of random bipartite planar maps with a prescribed degree sequence, Random Structures & Algorithms 53 (2018), no. 3, 448–503.  MR3854042

50.

Amram Meir and John W. Moon, On the maximum out-degree in random trees, Australas. J. Comb. 2 (1990), 147–156.  MR1126991Amram Meir and John W. Moon, On the maximum out-degree in random trees, Australas. J. Comb. 2 (1990), 147–156.  MR1126991

51.

Grégory Miermont, Tessellations of random maps of arbitrary genus, Annales Scientifiques de l’Ecole Normale Superieure 42 (2007).  MR2571957Grégory Miermont, Tessellations of random maps of arbitrary genus, Annales Scientifiques de l’Ecole Normale Superieure 42 (2007).  MR2571957

52.

Grégory Miermont, The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Mathematica 210 (2013), no. 2, 319–401.  MR3204495Grégory Miermont, The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Mathematica 210 (2013), no. 2, 319–401.  MR3204495

53.

Konstantinos Panagiotou, Benedikt Stufler, and Kerstin Weller, Scaling limits of random graphs from subcritical classes, The Annals of Probability 44 (2016), no. 5, 3291–3334.  MR3551197Konstantinos Panagiotou, Benedikt Stufler, and Kerstin Weller, Scaling limits of random graphs from subcritical classes, The Annals of Probability 44 (2016), no. 5, 3291–3334.  MR3551197

54.

J. Pitman, Combinatorial stochastic processes, Lecture Notes in Mathematics, vol. 1875, Springer-Verlag, Berlin, 2006, Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard.  MR2245368J. Pitman, Combinatorial stochastic processes, Lecture Notes in Mathematics, vol. 1875, Springer-Verlag, Berlin, 2006, Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard.  MR2245368

55.

Loïc Richier, Limits of the boundary of random planar maps, Probability Theory and Related Fields 172 (2018), no. 3, 789–827.  MR3877547Loïc Richier, Limits of the boundary of random planar maps, Probability Theory and Related Fields 172 (2018), no. 3, 789–827.  MR3877547

56.

L. Bruce Richmond and Nicholas C. Wormald, Almost all maps are asymmetric, J. Comb. Theory, Ser. B 63 (1995), 1–7.  MR1309353L. Bruce Richmond and Nicholas C. Wormald, Almost all maps are asymmetric, J. Comb. Theory, Ser. B 63 (1995), 1–7.  MR1309353

57.

Delphin Sénizergues, Sigurdur Örn Stefánsson, and Benedikt Stufler, Decorated stable trees, arXiv: 2205.02968 (2022).  MR4669752Delphin Sénizergues, Sigurdur Örn Stefánsson, and Benedikt Stufler, Decorated stable trees, arXiv: 2205.02968 (2022).  MR4669752

58.

Sigurdur Örn Stefánsson and Benedikt Stufler, Geometry of large Boltzmann outerplanar maps, Random Structures & Algorithms 55 (2019), no. 3, 742–771.  MR3997486Sigurdur Örn Stefánsson and Benedikt Stufler, Geometry of large Boltzmann outerplanar maps, Random Structures & Algorithms 55 (2019), no. 3, 742–771.  MR3997486

59.

Benedikt Stufler, Limits of random tree-like discrete structures, Probability Surveys 17 (2020), 318–477.  MR4132643Benedikt Stufler, Limits of random tree-like discrete structures, Probability Surveys 17 (2020), 318–477.  MR4132643

60.

W. T. Tutte, A census of slicings, Canadian Journal of Mathematics 14 (1962), 708–722.  MR0142470W. T. Tutte, A census of slicings, Canadian Journal of Mathematics 14 (1962), 708–722.  MR0142470

61.

W. T. Tutte, A census of planar maps, Canadian Journal of Mathematics 15 (1963), 249–271.  MR0146823W. T. Tutte, A census of planar maps, Canadian Journal of Mathematics 15 (1963), 249–271.  MR0146823
William Fleurat and Zéphyr Salvy "A phase transition in block-weighted random maps," Electronic Journal of Probability 29(none), 1-61, (2024). https://doi.org/10.1214/24-EJP1089
Received: 11 April 2023; Accepted: 20 January 2024; Published: 2024
Vol.29 • 2024
Back to Top