Electronic Communications in Probability

The Brownian continuum random tree as the unique solution to a fixed point equation

Marie Albenque and Christina Goldschmidt

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In this note, we provide a new characterization of Aldous' Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation).  We also show that this fixed point is attractive.

Article information

Electron. Commun. Probab. Volume 20 (2015), paper no. 61, 14 pp.

Accepted: 8 September 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05C05: Trees

continuum random tree fixed point equation

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Albenque, Marie; Goldschmidt, Christina. The Brownian continuum random tree as the unique solution to a fixed point equation. Electron. Commun. Probab. 20 (2015), paper no. 61, 14 pp. doi:10.1214/ECP.v20-4250. https://projecteuclid.org/euclid.ecp/1465320988

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  • Abraham, Romain; Delmas, Jean-François; Hoscheit, Patrick. A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18 (2013), no. 14, 21 pp.
  • Addario-Berry, L.; Broutin, N.; Goldschmidt, C. Critical random graphs: limiting constructions and distributional properties. Electron. J. Probab. 15 (2010), no. 25, 741–775.
  • Aldous, David. The continuum random tree. I. Ann. Probab. 19 (1991), no. 1, 1–28.
  • Aldous, David. The continuum random tree. II. An overview, Stochastic analysis (Durham, 1990), London Math. Soc. Lecture Note Ser., vol. 167, Cambridge University Press, Cambridge, 1991, pp. 23–70.
  • Aldous, David. The continuum random tree. III. Ann. Probab. 21 (1993), no. 1, 248–289.
  • Aldous, David. Recursive self-similarity for random trees, random triangulations and Brownian excursion. Ann. Probab. 22 (1994), no. 2, 527–545.
  • Bertoin, Jean; Miermont, Grégory. The cut-tree of large Galton-Watson trees and the Brownian CRT. Ann. Appl. Probab. 23 (2013), no. 4, 1469–1493.
  • Broutin, Nicolas; Marckert, Jean-François. Asymptotics of trees with a prescribed degree sequence and applications. Random Structures Algorithms 44 (2014), no. 3, 290–316.
  • Croydon, David; Hambly, Ben. Self-similarity and spectral asymptotics for the continuum random tree. Stochastic Process. Appl. 118 (2008), no. 5, 730–754.
  • N. Curien, B. Haas, and I. Kortchemski, The CRT is the scaling limit of random dissections, arXiv:1305.3534 [math.PR], 2013.
  • Durrett, Richard; Liggett, Thomas M. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 275–301.
  • Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 (2009), no. 1-2, 285–322.
  • Haas, Bénédicte; Miermont, Grégory. The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 (2004), no. 4, 57–97 (electronic).
  • Haas, Bénédicte; Miermont, Grégory. Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees. Ann. Probab. 40 (2012), no. 6, 2589–2666.
  • Le Gall, Jean-François. Random trees and applications. Probab. Surv. 2 (2005), 245–311.
  • Le Gall, Jean-François. Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 1, 35–62.
  • Marckert, Jean-François; Miermont, Grégory. The CRT is the scaling limit of unordered binary trees. Random Structures Algorithms 38 (2011), no. 4, 467–501.
  • Miermont, Grégory. Invariance principles for spatial multitype Galton-Watson trees. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 6, 1128–1161.
  • Neininger, Ralph; Rüschendorf, Ludger. A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 (2004), no. 1, 378–418.
  • Neininger, Ralph; Sulzbach, Henning. On a functional contraction method. Ann. Probab. 43 (2015), no. 4, 1777–1822.
  • Panagiotou, K.; Stufler, B.; Weller, K. Scaling limits of random graphs from subcritical classes, arXiv preprint arXiv:1411.1865 (2014).
  • Rösler, Uwe. A fixed point theorem for distributions. Stochastic Process. Appl. 42 (1992), no. 2, 195–214.
  • Rösler, U.; Rüschendorf, L. The contraction method for recursive algorithms. Average-case analysis of algorithms (Princeton, NJ, 1998). Algorithmica 29 (2001), no. 1-2, 3–33.
  • Benedikt Stufler, The continuum random tree is the scaling limit of unlabelled unrooted trees, arXiv preprint arXiv:1412.6333 (2014).