一维递阶合并过程的普适性
Universality in one-dimensional hierarchical coalescence processes
摘要：
受物理学文献中介绍的研究一维系统非平衡粗化动力学的几种模型的启发，我们考虑了一大类“分级聚结过程”（HCP）。HCP由一个无穷序列的聚结过程${ξ^{(n)}(\cdot)}_{n\ge1}$组成：每个过程发生在一个不同的“时期”（由$n$索引），并在无限的时间内演化，而随后时期的演化以这样一种方式联系在一起，即$ξ^{(n+1)}$的初始分布与$ξ^{(n)}$的最终分布一致。在每一个时期内的过程，由一个合适的简单的点过程表示相邻区间（域）之间的边界，演变如下。只有其长度属于某个与时代相关的有限范围的区间才是活跃的，也就是说，它们可以以相当一般的速率合并它们的左或右相邻区间。非活动间隔无法合并其邻居，并且只有在被活动邻居合并时才能增加其长度。活动范围是这样的，在合并步骤之后，新产生的间隔对于该时期总是变得不活动，但是对于某个未来时期是活动的。在不做任何平均场假设的情况下，我们证明了：（i）如果初始分布描述了一个更新过程，那么这样的属性在所有以后的时间和所有未来的时期都被保留;（ii）某些重新标度的变量的分布，例如，域长度，具有定义明确的和普遍的极限行为，作为独立于过程的细节（合并率，活动范围$,...$）的$n\to \infty$。最后一个结果解释了几个非常不同的物理系统（例如，玻璃态动力学的East模型或Paste-all模型），其在若干模拟中被观察到并在许多物理论文中被分析。获得渐近结果的主要思想是，首先写出一组递归的非线性恒等式，用于不同时期的相关量的拉普拉斯变换，然后通过在某种意义上使系统线性化的变换来求解。
英文摘要：
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 ${ξ^{(n)}(\cdot)}_{n\ge1}$: 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 $ξ^{(n+1)}$ coincides with the final distribution of $ξ^{(n)}$. 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.
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