## Bernoulli

• Bernoulli
• Volume 25, Number 3 (2019), 2301-2329.

### Gromov–Hausdorff–Prokhorov convergence of vertex cut-trees of $n$-leaf Galton–Watson trees

#### Abstract

In this paper, we study the vertex cut-trees of Galton–Watson trees conditioned to have $n$ leaves. This notion is a slight variation of Dieuleveut’s vertex cut-tree of Galton–Watson trees conditioned to have $n$ vertices. Our main result is a joint Gromov–Hausdorff–Prokhorov convergence in the finite variance case of the Galton–Watson tree and its vertex cut-tree to Bertoin and Miermont’s joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut’s and Bertoin and Miermont’s Gromov–Prokhorov convergence to Gromov–Hausdorff–Prokhorov remains open for their models conditioned to have $n$ vertices.

#### Article information

Source
Bernoulli, Volume 25, Number 3 (2019), 2301-2329.

Dates
Revised: June 2018
First available in Project Euclid: 12 June 2019

https://projecteuclid.org/euclid.bj/1560326446

Digital Object Identifier
doi:10.3150/18-BEJ1055

Mathematical Reviews number (MathSciNet)
MR3961249

Zentralblatt MATH identifier
07066258

#### Citation

He, Hui; Winkel, Matthias. Gromov–Hausdorff–Prokhorov convergence of vertex cut-trees of $n$-leaf Galton–Watson trees. Bernoulli 25 (2019), no. 3, 2301--2329. doi:10.3150/18-BEJ1055. https://projecteuclid.org/euclid.bj/1560326446

#### References

• [1] Abraham, R. and Delmas, J.-F. (2012). Record process on the continuum random tree. Ann. Probab. 40 1167–1211.
• [2] Abraham, R. and Delmas, J.-F. (2014). Local limits of conditioned Galton–Watson trees: The condensation case. Electron. J. Probab. 19 no. 56, 29.
• [3] Abraham, R., Delmas, J.-F. and He, H. (2012). Pruning Galton–Watson trees and tree-valued Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 48 688–705.
• [4] Abraham, R., Delmas, J.-F. and Hoscheit, P. (2013). A note on the Gromov–Hausdorff–Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18 no. 14, 21.
• [5] Addario-Berry, L., Broutin, N. and Holmgren, C. (2014). Cutting down trees with a Markov chainsaw. Ann. Appl. Probab. 24 2297–2339.
• [6] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
• [7] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
• [8] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
• [9] Aldous, D. and Pitman, J. (1998). Tree-valued Markov chains derived from Galton–Watson processes. Ann. Inst. Henri Poincaré Probab. Stat. 34 637–686.
• [10] Aldous, D. and Pitman, J. (2000). Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Probab. Theory Related Fields 118 455–482.
• [11] Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. Henri Poincaré Probab. Stat. 38 319–340.
• [12] Bertoin, J. (2012). Fires on trees. Ann. Inst. Henri Poincaré Probab. Stat. 48 909–921.
• [13] Bertoin, J. and Miermont, G. (2013). The cut-tree of large Galton–Watson trees and the Brownian CRT. Ann. Appl. Probab. 23 1469–1493.
• [14] Broutin, N. and Wang, M. (2017). Cutting down ${\mathbf{p}}$-trees and inhomogeneous continuum random trees. Bernoulli 23 2380–2433.
• [15] Camarri, M. and Pitman, J. (2000). Limit distributions and random trees derived from the birthday problem with unequal probabilities. Electron. J. Probab. 5 no. 2, 18.
• [16] de Raphélis, L. (2017). Scaling limit of multitype Galton–Watson trees with infinitely many types. Ann. Inst. Henri Poincaré Probab. Stat. 53 200–225.
• [17] Dieuleveut, D. (2015). The vertex-cut-tree of Galton–Watson trees converging to a stable tree. Ann. Appl. Probab. 25 2215–2262.
• [18] Duquesne, T. (2003). A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31 996–1027.
• [19] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147.
• [20] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
• [21] Duquesne, T. and Winkel, M. (2012). Hereditary tree growth and Lévy forests. Stoch. Proc. Appl. To appear. Available at arXiv:1211.2179.
• [22] Duquesne, T. and Winkel, M. (2017). Hereditary tree growth and decompositions. In preparation.
• [23] Evans, S.N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81–126.
• [24] Evans, S.N. and Winter, A. (2006). Subtree prune and regraft: A reversible real tree-valued Markov process. Ann. Probab. 34 918–961.
• [25] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II. 2nd ed. New York–London–Sydney: Wiley.
• [26] Greven, A., Pfaffelhuber, P. and Winter, A. (2009). Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 285–322.
• [27] Haas, B. and Miermont, G. (2004). The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 57–97.
• [28] Haas, B. and Miermont, G. (2012). Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 2589–2666.
• [29] He, H. and Winkel, M. (2014). Invariance principles for pruning processes of Galton–Watson trees. Available at arXiv:1409.1014.
• [30] Ibragimov, I.A. and Linnik, Yu.V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff Publishing.
• [31] Janson, S. (2006). Random cutting and records in deterministic and random trees. Random Structures Algorithms 29 139–179.
• [32] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). New York: Springer.
• [33] Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22 425–487.
• [34] Kortchemski, I. (2012). Invariance principles for Galton–Watson trees conditioned on the number of leaves. Stochastic Process. Appl. 122 3126–3172.
• [35] Kortchemski, I. (2014). Random stable laminations of the disk. Ann. Probab. 42 725–759.
• [36] Meir, A. and Moon, J.W. (1970). Cutting down random trees. J. Aust. Math. Soc. 11 313–324.
• [37] Miermont, G. (2008). Invariance principles for spatial multitype Galton–Watson trees. Ann. Inst. Henri Poincaré Probab. Stat. 44 1128–1161.
• [38] Miermont, G. (2009). Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 725–781.
• [39] Neveu, J. (1986). Erasing a branching tree. Adv. in Appl. Probab. 18 101–108.
• [40] Neveu, J. and Pitman, J. (1989). Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. In Séminaire de Probabilités, XXIII. Lecture Notes in Math. 1372 239–247. Springer, Berlin.
• [41] Neveu, J. and Pitman, J.W. (1989). The branching process in a Brownian excursion. In Séminaire de Probabilités, XXIII. Lecture Notes in Math. 1372 248–257. Springer, Berlin.
• [42] Panholzer, A. (2006). Cutting down very simple trees. Quaest. Math. 29 211–227.
• [43] Pitman, J. (1999). Coalescent random forests. J. Combin. Theory Ser. A 85 165–193.
• [44] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Berlin: Springer.
• [45] Pitman, J. and Rizzolo, D. (2015). Schröder’s problems and scaling limits of random trees. Trans. Amer. Math. Soc. 367 6943–6969.
• [46] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Berlin: Springer.
• [47] Rizzolo, D. (2015). Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set. Ann. Inst. Henri Poincaré Probab. Stat. 51 512–532.