The Annals of Applied Probability

Coagulation–fragmentation duality, Poisson–Dirichlet distributions and random recursive trees

Rui Dong, Christina Goldschmidt, and James B. Martin

Full-text: Open access


In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson–Dirichlet distributions PD (α,θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Fragα and Coagα,θ, respectively, with the following property: if the input to Fragα has PD (α,θ) distribution, then the output has PD (α,θ+1) distribution, while the reverse is true for Coagα,θ. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD (α,θ) and PD (αβ,θ). Repeated application of the Fragα operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality.

Article information

Ann. Appl. Probab., Volume 16, Number 4 (2006), 1733-1750.

First available in Project Euclid: 17 January 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G51: Processes with independent increments; Lévy processes 60C05: Combinatorial probability 05C05: Trees

Coagulation fragmentation Poisson–Dirichlet distributions recursive trees time reversal


Dong, Rui; Goldschmidt, Christina; Martin, James B. Coagulation–fragmentation duality, Poisson–Dirichlet distributions and random recursive trees. Ann. Appl. Probab. 16 (2006), no. 4, 1733--1750. doi:10.1214/105051606000000655.

Export citation


  • Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1--28.
  • Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Math. Soc. Lecture Note Ser. 167 23--70. Cambridge Univ. Press.
  • Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248--289.
  • Aldous, D. and Pitman, J. (1998). The standard additive coalescent. Ann. Probab. 26 1703--1726.
  • Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509--512.
  • Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 319--340.
  • Bertoin, J. and Goldschmidt, C. (2004). Dual random fragmentation and coagulation and an application to the genealogy of Yule processes. In Mathematics and Computer Science III (Vienna 2004). Trends Math. 295--308. Birkhäuser, Basel.
  • Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353--355.
  • Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18 279--290.
  • Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247--276.
  • Dynkin, E. B. (1965). Markov Processes. I. Academic Press, New York.
  • Haas, B., Miermont, G., Pitman, J. and Winkel, M. (2005). Asymptotics of discrete fragmentation trees and applications to phylogenetic models. Available at
  • Kingman, J. F. C. (1975). Random discrete distributions. J. Roy. Statist. Soc. Ser. B 37 1--22.
  • Mahmoud, H. M., Smythe, R. T. and Szymański, J. (1993). On the structure of random plane-oriented recursive trees and their branches. Random Structures Algorithms 4 151--176.
  • Móri, T. F. (2002). On random trees. Studia Sci. Math. Hungar. 39 143--155.
  • Pitman, J. (1992). Notes on the two parameter generalization of Ewens' random partition structure. Unpublished manuscript.
  • Pitman, J. (1996). Some developments of the Blackwell--MacQueen urn scheme. In Statistics, Probability and Game Theory (T. S. Ferguson, L. S. Shapley and J. B. MacQueen, eds.) 245--267. IMS, Hayward, CA.
  • Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870--1902.
  • Pitman, J. (2006). Combinatorial Stochastic Processes. Springer, Berlin.
  • Pitman, J. and Yor, M. (1997). The two-parameter Poisson--Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855--900.
  • Rudas, A., Tóth, B. and Valkó, B. (2006). Random trees and general branching processes. Random Structures Algorithms. To appear. Available at
  • Smythe, R. T. and Mahmoud, H. M. (1994). A survey of recursive trees. Teor. Ĭmov\=\i r. Mat. Stat. (Theor. Probability and Math. Statist.) 51 1--29.
  • Szymański, J. (1987). On a nonuniform random recursive tree. In Random Graphs '85 (Poznań, 1985). North-Holland Math. Stud. 144 297--306. North-Holland, Amsterdam.