Universality in one-dimensional hierarchical coalescence processes

TL;DR

This paper studies hierarchical coalescence processes in one-dimensional systems, showing that their long-term behavior is universal across different models and initial conditions, with implications for understanding physical phenomena like glassy dynamics.

Contribution

It introduces a broad class of hierarchical coalescence processes and proves their universal limiting behavior, independent of specific model details.

Findings

Rescaled domain length distributions converge to a universal limit.

Renewal process initial conditions are preserved over time.

The results explain observed universality in physical systems like glassy dynamics.

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 ${\xi^{(n)}(\cdot)}_{n\ge1}$: 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$,...$). 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.