Universality for onedimensional hierarchical coalescence processes with double and triple merges
Abstract
We consider onedimensional 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 epochdependent 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) 13771435] to a larger family of models, including relevant examples from the physics literature [Europhys. Lett. 27 (1994) 175180, 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2011
 DOI:
 10.48550/arXiv.1112.4630
 arXiv:
 arXiv:1112.4630
 Bibcode:
 2011arXiv1112.4630F
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Probability
 EPrint:
 Published in at http://dx.doi.org/10.1214/12AAP917 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)