  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.
