加速有向蠕虫算法的几何分配方法
Geometric allocation approach to accelerating directed worm algorithm
摘要：
蠕虫算法是马尔可夫链蒙特卡罗方法中的一种通用技术，适用于经典和量子系统。该算法大大简化了临界慢化，降低了各种经典系统的动态临界指数。因此，改进蒙特卡罗方法的算法，拓展蒙特卡罗方法在物理系统中的应用范围是十分必要的。我们在这里提出了一个有向蠕虫算法，显着提高计算效率。我们使用几何分配的方法来优化蠕虫的散射过程：避免了蠕虫的后向散射，前向散射是有利的。我们的方法成功地提高了扩散的蠕虫头（扭结），这是显而易见的两个扭结的相对位置的概率分布。通过测量指数自相关时间和渐近方差，证明了伊辛模型在临界温度下的性能改善。目前的蠕虫更新是大约25倍的效率，为简单的立方格子模型的传统蠕虫更新。令人惊讶的是，我们的算法甚至比Wolff聚类算法更有效，这是最好的更新算法之一。我们估计简单立方格子伊辛模型的动态临界指数为$z \approx 0.27$在蠕虫更新。蠕虫和沃尔夫算法产生不同的指数的综合自相关时间的磁化率估计，但相同的指数的渐近方差。我们还讨论了如何量化马尔可夫链蒙特卡罗方法的计算效率。我们的方法可以应用于广泛的物理系统，如$| ϕ|^4$模型，Potts模型，O（$n$）圈模型，和格点量子色动力学。
英文摘要：
The worm algorithm is a versatile technique in the Markov chain Monte Carlo method for both classical and quantum systems. The algorithm substantially alleviates critical slowing down and reduces the dynamic critical exponents of various classical systems. It is crucial to improve the algorithm and push the boundary of the Monte Carlo method for physical systems. We here propose a directed worm algorithm that significantly improves computational efficiency. We use the geometric allocation approach to optimize the worm scattering process: worm backscattering is averted, and forward scattering is favored. Our approach successfully enhances the diffusivity of the worm head (kink), which is evident in the probability distribution of the relative position of the two kinks. Performance improvement is demonstrated for the Ising model at the critical temperature by measurement of exponential autocorrelation times and asymptotic variances. The present worm update is approximately 25 times as efficient as the conventional worm update for the simple cubic lattice model. Surprisingly, our algorithm is even more efficient than the Wolff cluster algorithm, which is one of the best update algorithms. We estimate the dynamic critical exponent of the simple cubic lattice Ising model to be $z \approx 0.27$ in the worm update. The worm and the Wolff algorithms produce different exponents of the integrated autocorrelation time of the magnetic susceptibility estimator but the same exponent of the asymptotic variance. We also discuss how to quantify the computational efficiency of the Markov chain Monte Carlo method. Our approach can be applied to a wide range of physical systems, such as the $| ϕ|^4$ model, the Potts model, the O($n$) loop model, and lattice QCD.
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