标题： 极限大偏差率函数的数值估计 显示英文标题

标题： Numerical Estimation of Limiting Large-Deviation Rate Functions

摘要： 对于遵循大偏差原理的系统中稀有事件的统计，速率函数是一个关键量。 当数值估计速率函数时，总是受到有限系统大小的限制。 因此，如果关注无限系统大小的极限速率函数，首先需要数值研究多个系统大小。 在此，使用偏差系综的稀有事件算法可以访问低概率区域。 其次，必须进行某种系统大小的外推。 这里我们展示如何将稀有事件重要性采样方案与多直方图重加权相结合，这使得该方法具有相当广泛的适用性，而无需特定采样算法。 我们研究了两种进行系统大小外推的方法，一种是直接作用于经验速率函数，另一种是作用于标度累积生成函数，以获得无限大小的极限。 所提出的方法针对二项分布变量和Erdös-Rényi随机图中的最大连通组件进行了演示。 在这两种情况下都有解析解可供直接比较。 特别是观察到，在偏差系综中出现的相变可能导致与真实结果的系统性偏差。 显示英文摘要

摘要： For statistics of rare events in systems obeying a large-deviation principle, the rate function is a key quantity. When numerically estimating the rate function one is always restricted to finite system sizes. Thus, if the interest is in the limiting rate function for infinite system sizes, first, several system sizes have to be studied numerically. Here, rare-event algorithms using biased ensembles give access to the low-probability region. Second, some kind of system-size extrapolation has to be performed. Here we demonstrate how rare-event importance sampling schemes can be combined with multi-histogram reweighting, which allows for rather general applicability of the approach, independent of specific sampling algorithms. We study two ways of performing the system-size extrapolation, either directly acting on the empirical rate functions, or on the scaled cumulant generating functions, to obtain the infinite-size limit. The presented method is demonstrated for a binomial distributed variable and the largest connected component in Erd\"os-R\'enyi random graphs. Analytical solutions are available in both cases for direct comparison. It is observed in particular that phase transitions appearing in the biased ensembles can lead to systematic deviations from the true result.