Universality in one dimensional Hierarchical Coalescence Processes
Abstract
Motivated by several models introduced in the physical literature to study the non-equilibrium 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 $\{\xi^{(n)}(\cdot)\}_{n\ge 1}$: 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, i.e. they can incorporate their left or right neighboring interval with quite general rates. Inactive intervals cannot incorporate their neighbours and can increase their length only if they are incorporated by active neighbours. 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, e.g. the domain length, has a well defined and universal limiting behavior as $n\to \infty$ independent of the details of the process. 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 Past-all-model) which was observed in several simulations and analyzed in many physical papers.