The Annals of Applied Probability

Universality for one-dimensional hierarchical coalescence processes with double and triple merges

A. Faggionato, C. Roberto, and C. Toninelli

Full-text: Open access


We consider one-dimensional hierarchical coalescence processes (in short HCPs) where two or three neighboring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different “epoch” and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighboring domains or both of them if its length belongs to a certain epoch-dependent finite range.

Assuming that the distribution at the beginning of the first epoch is described by a renewal simple point process, we prove limit theorems for the domain length and for the position of the leftmost point (if any). Our analysis extends the results obtained in [Ann. Probab. 40 (2012) 1377–1435] to a larger family of models, including relevant examples from the physics literature [Europhys. Lett. 27 (1994) 175–180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing us to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation.

Article information

Ann. Appl. Probab., Volume 24, Number 2 (2014), 476-525.

First available in Project Euclid: 10 March 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes 60B10: Convergence of probability measures

Coalescence process simple point process renewal process universality nonequilibrium dynamics


Faggionato, A.; Roberto, C.; Toninelli, C. Universality for one-dimensional hierarchical coalescence processes with double and triple merges. Ann. Appl. Probab. 24 (2014), no. 2, 476--525. doi:10.1214/12-AAP917.

Export citation


  • [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [2] Bray, A. J., Derrida, B. and Gordrèche, C. (1994). Nontrivial algebraic decay in a soluble model of coarsening. Europhys. Lett. 27 175–180.
  • [3] Carr, J. and Pego, R. (1992). Self-similarity in a coarsening model in one dimension. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 436 569–583.
  • [4] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • [5] Derrida, B., Bray, A. J. and Godrèche, C. (1994). Nontrivial exponents in the zero temperature dynamics of the 1d Ising and Potts model. J. Phys. A 27 L357–L361.
  • [6] Derrida, B., Godrèche, C. and Yekutieli, I. (1990). Stable distributions of growing and coalescing droplets. Europhys. Lett. 12 385–390.
  • [7] Derrida, B., Godrèche, C. and Yekutieli, I. (1991). Scale invariant regime in the one dimensional models of growing and coalescing droplets. Phys. Rev. A (3) 44 6241–6251.
  • [8] Eisinger, S. and Jackle, J. (1991). A hierarchically constrained kinetic ising model. Z. Phys. B 84 115–124.
  • [9] Faggionato, A., Martinelli, F., Roberto, C. and Toninelli, C. (2012). Universality in one-dimensional hierarchical coalescence processes. Ann. Probab. 40 1377–1435.
  • [10] Faggionato, A., Martinelli, F., Roberto, C. and Toninelli, C. (2012). Aging through hierarchical coalescence in the East model. Comm. Math. Phys. 309 459–495.
  • [11] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, 2nd ed. Wiley Series in Probability and Mathematical Statistics 2. Wiley, New York.
  • [12] Franken, P., König, D., Arndt, U. and Schmidt, V. (1982). Queues and Point Processes. Wiley, Chichester.
  • [13] Gallay, T. and Mielke, A. (2003). Convergence results for a coarsening model using global linearization. J. Nonlinear Sci. 13 311–346.
  • [14] Garcia, N. L. and Kurtz, T. G. (2006). Spatial birth and death processes as solutions of stochastic equations. ALEA Lat. Am. J. Probab. Math. Stat. 1 281–303.
  • [15] Liggett, T. M. (2005). Interacting Particle Systems. Grundlehren der mathematischen Wissenschaften 276. Springer, Berlin.
  • [16] Preston, C. (1975). Spatial birth-and-death processes. In Proceedings of the 40th Session of the International Statistical Institute (Warsaw, 1975), Vol. 2. Invited Papers 46 371–391, 405–408.
  • [17] Privman, V. (1997). Nonequilibrium Statistical Physics in One Dimension. Cambridge Univ. Press, Cambridge.
  • [18] Seppäläinen, T. Translation invariant exclusion processes. Available at
  • [19] Sollich, P. and Evans, M. R. (2003). Glassy dynamics in the asymmetrically constrained kinetic Ising chain. Phys. Rev. E (3) 68 031504.