再谈欧几里得动态三角剖分：相变真的是一阶相变吗？（扩展版本）
Euclidean Dynamical Triangulation revisited: is the phase transition really 1st order? (extended version)
摘要：
4D欧几里得动态三角测量[1]的两个相位之间的过渡一直被认为是二阶的，直到1996年发现足够大的系统的一阶行为[5，9]。然而，人们可能会想知道这一发现是否受到所使用的数值方法的影响：为了控制音量波动，在两项研究中[5，9]，人为谐波电位被添加到动作中;在[9]中，在固定数量的接受而不是尝试移动后进行测量，这引入了额外的误差。最后，模拟遭受强烈的临界减速，这可能被低估了。在目前的工作中，我们解决了上述弱点：我们允许体积在固定的时间间隔内自由波动;我们在固定数量的尝试移动后进行测量;我们通过使用优化的并行回火算法克服了临界减速[12]。用这些改进的方法，在大小为64 k的4-单形系统上，我们证实了相变是一级的。此外，我们讨论了一个局部的标准，以决定是否部分的三角形是在伸长或皱巴巴的状态，并描述了一个新的对应EDT和球的盒子模型。后者产生了一个修改后的配分函数与一个额外的，第三耦合。最后，我们提出并激发了一类修正的路径积分措施，可能会删除亚稳态的马尔可夫链，把相变到二阶。
英文摘要：
The transition between the two phases of 4D Euclidean Dynamical Triangulation [1] was long believed to be of second order until in 1996 first order behavior was found for sufficiently large systems [5,9]. However, one may wonder if this finding was affected by the numerical methods used: to control volume fluctuations, in both studies [5,9] an artificial harmonic potential was added to the action; in [9] measurements were taken after a fixed number of accepted instead of attempted moves which introduces an additional error. Finally the simulations suffer from strong critical slowing down which may have been underestimated. In the present work, we address the above weaknesses: we allow the volume to fluctuate freely within a fixed interval; we take measurements after a fixed number of attempted moves; and we overcome critical slowing down by using an optimized parallel tempering algorithm [12]. With these improved methods, on systems of size up to 64k 4-simplices, we confirm that the phase transition is first order. In addition, we discuss a local criterion to decide whether parts of a triangulation are in the elongated or crumpled state and describe a new correspondence between EDT and the balls in boxes model. The latter gives rise to a modified partition function with an additional, third coupling. Finally, we propose and motivate a class of modified path-integral measures that might remove the metastability of the Markov chain and turn the phase transition into second order.
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