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

TL;DR

This paper studies one-dimensional hierarchical coalescence processes involving double and triple merges, proving limit theorems and revealing a universal structure that connects various models from physics.

Contribution

It extends previous results to a broader class of models, characterizes the infinitesimal generator, and links different models through a common abstract framework.

Findings

Proves limit theorems for domain length and position of the leftmost point.

Identifies a universal abstract structure behind diverse models.

Provides a full characterization of the infinitesimal generator.

Abstract

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.