Universality in one-dimensional hierarchical coalescence processes. (English) Zbl 1264.60033

Faggionato, Alessandra; Martinelli, Fabio; Roberto, Cyril; Toninelli, Cristina

Ann. Probab. 40, No. 4, 1377-1435 (2012).

Authors’ abstract: Motivated by several models introduced in the physics literature to study the nonequilibrium coarsening dynamics of one-dimensional systems, we consider a large class of “hierarchical coalescence processes” (HCP). An HCP consists of an infinite sequence of coalescence processes \(\{\xi^{(n)}(\cdot)\}_{n\geq 1}\): each process occurs in a different “epoch” (indexed by \(n\)) and evolves for an infinite time, while the evolution in subsequent epochs are linked in such a way that the initial distribution of \(\xi^{(n+1)}\) coincides with the final distribution of \(\xi^{(n)}\). Inside each epoch the process, described by a suitable simple point process representing the boundaries between adjacent intervals (domains), evolves as follows. Only intervals whose length belongs to a certain epoch-dependent finite range are active, that is, they can incorporate their left or right neighboring interval with quite general rates. Inactive intervals cannot incorporate their neighbors and can increase their length only if they are incorporated by active neighbors. The activity ranges are such that after a merging step the newly produced interval always becomes inactive for that epoch but active for some future epoch.
Without making any mean-field assumption, we show that (i) if the initial distribution describes a renewal process, then such a property is preserved at all later times and all future epochs; (ii) the distribution of certain rescaled variables, for example, the domain length, has a well-defined and universal limiting behavior as \(n\to\infty\) independent of the details of the process (merging rates, activity ranges, etc.). This last result explains the universality in the limiting behavior of several very different physical systems (e.g., the East model of glassy dynamics or the Paste-all model) which was observed in several simulations and analyzed in many physics papers. The main idea to obtain the asymptotic result is to first write down a recursive set of nonlinear identities for the Laplace transforms of the relevant quantities on different epochs and then to solve it by means of a transformation which in some sense linearizes the system.

Reviewer: Wolfgang Freudenberg (Cottbus)

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60B10	Convergence of probability measures
82C24	Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
60G10	Stationary stochastic processes

Keywords:

coalescence process; simple point process; renewal process; universality; nonequlibrium dynamics

References:

[1]	Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3-48. · Zbl 0930.60096 · doi:10.2307/3318611
[2]	Bertoin, J. (2001). Eternal additive coalescents and certain bridges with exchangeable increments. Ann. Probab. 29 344-360. · Zbl 1019.60072 · doi:10.1214/aop/1008956333
[3]	Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102 . Cambridge Univ. Press, Cambridge. · Zbl 1107.60002 · doi:10.1017/CBO9780511617768
[4]	Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[5]	Bray, A. J., Derrida, B. and Gordrèche, C. (1994). Non-trivial algebraic decay in a soluble model of coarsening. Europhys. Lett. 27 175-180. · Zbl 0935.82016 · doi:10.1209/0295-5075/27/3/001
[6]	Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes . Springer, New York. · Zbl 0657.60069
[7]	Derrida, B., Bray, A. J. and Godrèche, C. (1994). Non-trivial exponents in the zero temperature dynamics of the 1d Ising and Potts model. J. Phys. A 27 L357-L361. · Zbl 0842.60093 · doi:10.1088/0305-4470/27/11/002
[8]	Derrida, B., Godrèche, C. and Yekutieli, I. (1990). Stable distributions of growing and coalescing droplets. Europhys. Lett. 12 385-390.
[9]	Derrida, B., Godrèche, C. and Yekutieli, I. (1991). Scale invariant regime in the one dimensional models of growing and coalescing droplets. Phys. Rev. A 44 6241-6251.
[10]	Durrett, R. (1996). Probability : Theory and Examples , 2nd ed. Duxbury Press, Belmont, CA. · Zbl 1202.60001
[11]	Eisinger, S. and Jackle, J. (1991). A hierarchically constrained kinetic ising model. Z. Phys. B 84 115-124.
[12]	Faggionato, A., Martinelli, F., Roberto, C. and Toninelli, C. (2010). Aging through hierarchical coalescence in the East model. Preprint. Available at . 1012.4912 · Zbl 1237.82029
[13]	Faggionato, A., Martinelli, F., Roberto, C. and Toninelli, C. (2011). Unpublished manuscript.
[14]	Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II , 2nd ed. Wiley, New York. · Zbl 0219.60003
[15]	Franken, P., König, D., Arndt, U. and Schmidt, V. (1982). Queues and Point Processes . Wiley, Chichester. · Zbl 0505.60058
[16]	Gallay, T. and Mielke, A. (2003). Convergence results for a coarsening model using global linearization. J. Nonlinear Sci. 13 311-346. · Zbl 1025.35006 · doi:10.1007/s00332-002-0543-8
[17]	Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles . Springer, New York. · Zbl 1084.60003
[18]	Limic, V. and Sturm, A. (2006). The spatial \(\Lambda \)-coalescent. Electron. J. Probab. 11 363-393 (electronic). · Zbl 1113.60077 · doi:10.1214/EJP.v11-319
[19]	Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes . Wiley, New York. · Zbl 0383.60001
[20]	Pego, R. L. (2007). Lectures on dynamics in models of coarsening and coagulation. In Dynamics in Models of Coarsening , Coagulation , Condensation and Quantization. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 9 1-61. World Scientific, Hackensack, NJ. · Zbl 1138.82021 · doi:10.1142/9789812770226_0001
[21]	Privman, V., ed. (1997). Non Equilibrium Statistical Mechanics in One Dimension . Cambridge Univ. Press, New York. · Zbl 0941.00045
[22]	Sollich, P. and Evans, M. R. (2003). Glassy dynamics in the asymmetrically constrained kinetic Ising chain. Phys. Rev. E 68 031504.