East模型中的等级聚合老化
Aging through hierarchical coalescence in the East model
摘要：
我们严格地分析了低温非平衡动力学的东方模型，一个特殊的例子，一个一维定向的动力学约束粒子模型，当初始分布是不同的可逆的和时间远小于全球弛豫时间。在物理学文献中已经深入研究了这种设置，以分析从液体到玻璃相的突然淬火之后的缓慢动力学。在零温度的限制（即一个消失的密度的空缺）和初始分布，使空缺形成一个更新过程，我们证明了空缺的密度，持久性函数和两次自相关函数表现为阶梯函数与几个高原。此外，两次自相关函数显示出老化行为。我们还提供了一个尖锐的描述统计的域长度作为时间的函数，域是两个连续的空缺之间的间隔。当初始更新过程具有有限平均值时，我们的结果证实（并推广）了物理学家以前对产品Bernoulli测度的限制情况下的发现。然而，我们发现，一个不同的行为出现时，初始域分布是在吸引域的$α$稳定的法律。所有上述结果实际上都来自一个更一般的结果，即East模型的低温动力学很好地描述了某个分层聚结过程的动力学，这是一个概率对象，可以被视为适当链接的聚结过程的分层序列，其渐近行为最近在[14]中进行了研究。
英文摘要：
We rigorously analyze the low temperature non-equilibrium dynamics of the East model, a special example of a one dimensional oriented kinetically constrained particle model, when the initial distribution is different from the reversible one and for times much smaller than the global relaxation time. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. In the limit of zero temperature (i.e. a vanishing density of vacancies) and for initial distributions such that the vacancies form a renewal process we prove that the density of vacancies, the persistence function and the two-time autocorrelation function behave as staircase functions with several plateaux. Furthermore the two-time autocorrelation function displays an aging behavior. We also provide a sharp description of the statistics of the domain length as a function of time, a domain being the interval between two consecutive vacancies. When the initial renewal process has finite mean our results confirm (and generalize) previous findings of the physicists for the restricted case of a product Bernoulli measure. However we show that a different behavior appears when the initial domain distribution is in the attraction domain of a $α$-stable law. All the above results actually follow from a more general result which says that the low temperature dynamics of the East model is very well described by that of a certain hierarchical coalescence process, a probabilistic object which can be viewed as a hierarchical sequence of suitably linked coalescence processes and whose asymptotic behavior has been recently studied in [14].
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