Universality in one-dimensional hierarchical coalescence processes

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.Comment: Published in at http://dx.doi.org/10.1214/11-AOP654 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Time Scale Separation and Dynamic Heterogeneity in the Low Temperature East Model

“…However the most interesting and challenging dynamical behavior, featuring aging and dynamic heterogeneity, occurs for q ≪ 1 on time scales shorter than the relaxation time. Building upon the non-rigorous picture in the physics literature [32] but going well beyond it, it was proved in [15] that, for all N independent of q, the dynamics of the infinite East chain in a space window [1, 2 N ] and up to time scales O(1/q N ) is well approximated, as q ↓ 0, by a certain hierarchical coalescence process (HCP) [17]. In this HCP vacancies are isolated and domains (maximal blocks of the form 111..10) with cardinality between 2 n−1 and 2 n , n N , coalesce with the domain at their right only at time scale ∼ (1/q) n .…”

Relaxation to equilibrium of generalized East processes on $\mathbb{Z}^{d}$: Renormalization group analysis and energy-entropy competition

“…[14], Theorems 2 and 5).Mathematically, the East model poses very challenging and interesting problems because of the hardness of the constraint and the fact that it is not attractive. It also has interesting ramifications in combinatorics [16], coalescence processes [19,20,22] and random walks on triangular matrices [32]. Moreover, some of the mathematical tools developed for the analysis of its relaxation time scales proved to be quite powerful also in other contexts such as card shuffling problems [5] and random evolution of surfaces [12].…”

A parity breaking Ising chain Hamiltonian as a Brownian motor

“…Since the 1980's such models have attracted the interest of physicists and mathematicians alike as models of glassy relaxation (see, e.g. [18,19,22,20,21,23,24]). Attention to the connection between kinetically constrained models and certain properties of Brownian molecular motors is only very recent [25,26].…”