Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality in a class of fragmentation-coalescence processes

A. E. Kyprianou, S. W. Pagett, and T. Rogers

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We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non-random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit a universal cluster size distribution regardless of the details of the rates, following a power law with exponent $3/2$.


Nous introduisons et étudions une classe de processus de fragmentation-coalescence définis sur des systèmes finis de particules organisées en amas. Lors d’un événement de coalescence, de multiples amas fusionnent simultanément en un amas plus gros, et dans le cas d’un événement de fragmentation, un amas est détruit en une collection de singletons. Sous des hypothèses faibles sur les taux de coalescence, nous montrons que la loi des tailles des amas devient déterministe dans la limite thermodynamique. De plus, nous montrons que dans la limite des petits taux de fragmentation, ces processus admettent une loi universelle pour la les tailles des amas, indépendante de la valeur précise des taux, et qui suit une loi de puissance avec un exposant $3/2$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 1134-1151.

Received: 23 December 2015
Revised: 30 March 2017
Accepted: 30 March 2017
First available in Project Euclid: 25 April 2018

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Zentralblatt MATH identifier

Primary: 60G99: None of the above, but in this section

Fragmentation-coalescence Cluster distribution Thermodynamic limit


Kyprianou, A. E.; Pagett, S. W.; Rogers, T. Universality in a class of fragmentation-coalescence processes. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 1134--1151. doi:10.1214/17-AIHP834.

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  • [1] D. J. Aldous. Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 (1) (1999) 3–48.
  • [2] D. Berend and T. Tassa. Improved bounds on Bell numbers and on moments of sums of random variables. Probab. Math. Statist. 30 (2) (2010) 185–205.
  • [3] J. Berestycki. Exchangeable fragmentation-coalescence processes and their equilibrium measures. Electron. J. Probab. 9 (25) (2004) 770–824.
  • [4] J. C. Bohorquez, S. Gourley, A. R. Dixon, M. Spagat and N. F. Johnson. Common ecology quantifies human insurgency. Nature 462 (7275) (2009) 911–914.
  • [5] X. Bressaud and N. Fournier. A mean-field forest-fire model. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2) (2014) 589–614.
  • [6] M. Eibl, E. Hendriks, J. Spouge and M. Schreckenberg. Exact solutions for random coagulation processes. Physik Z. 58 (3) (1985) 219–227.
  • [7] S. Gueron. The steady-state distributions of coagulation-fragmentation processes. J. Math. Biol. 37 (1) (1998) 1–27.
  • [8] S. Gueron and S. A. Levin. The dynamics of group formation. Math. Biosci. 128 (1) (1995) 243–264.
  • [9] M. Hazewinkel. Encyclopaedia of Mathematics, Vol. 5. Springer, Netherlands, 1990.
  • [10] H. Hinrichsen. Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49 (7) (2000) 815–958.
  • [11] I. Jeon. Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phys. 194 (3) (1998) 541–567.
  • [12] K. Koutzenogii, A. Levykin and K. Sabelfeld. Kinetics of aerosol formation in the free molecule regime in presence of condensable vapor. J. Aerosol Sci. 27 (5) (1996) 665–679.
  • [13] A. Lambert. The branching process with logistic growth. Ann. Appl. Probab. 15 (2) (2005) 1506–1535.
  • [14] A. Lushnikov. Certain new aspects of the coagulation theory. Izv. An. Fiz. Atmos. Phys. 14 (10) (1978) 738–743.
  • [15] A. Lushnikov. Coagulation in finite systems. J. Colloid. Interf. Sci. 65 (2) (1978) 276–285.
  • [16] A. H. Marcus. Stochastic coalescence. Technometrics 10 (1968) 133–143.
  • [17] B. Ráth and B. Tóth. Erdős–Rényi random graphs $+$ forest fires $=$ self-organized criticality. Electron. J. Probab. 14 (45) (2009) 1290–1327.
  • [18] J. Seinfeld. Atmospheric Chemistry and Physics of Air Pollution. Wiley, New York, 1986.
  • [19] M. Smoluchowski. Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Physik Z. 17 (1916) 557–585.
  • [20] S. Tavaré. Line-of-descent and genealogical processes, and their applications in population genetics models. Theor. Popul. Biol. 26 (2) (1984) 119–164.
  • [21] G. Teschl. Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics 140. American Mathematical Society, Providence, RI, 2012.
  • [22] F. Verhulst. Nonlinear Differential Equations and Dynamical Systems. Universitext. Springer, Berlin, 1990. Translated from the Dutch.