NSPT计算中的大波动：来自O(N)非线性sigma模型的启示
Large fluctuations in NSPT computations: a lesson from $O(N)$ non-linear sigma models
AI总结：
本文通过模拟二维O(N)非线性sigma模型，证明NSPT计算中的统计波动在大N极限下被抑制，从而可将微扰计算推进到高阶，并建立了场论方差与NSPT随机过程方差及协方差之间的一般关系。
摘要：
在过去三十年中，数值随机微扰理论（NSPT）已被证明是计算诸如格点QCD等理论中微扰展开的优秀工具，而对于这些理论，标准的图形微扰理论已知十分繁琐。尽管这种随机方法取得了显著成功，并且近年来有所改进，但NSPT显然无法在低维模型中成功实现，原因是出现了巨大的统计波动：随着微扰阶数升高，信噪比根本不够好。这并不令人意外，但从非常一般的角度来看，人们会预期自由度数目越大，波动就越不严重。通过模拟不同N值的二维O(N)非线性sigma模型，我们表明波动确实在大N极限下被抑制，符合我们的预期：对于大量内部自由度（即N足够大），NSPT微扰计算可以推进到很高的微扰阶数n。通过将我们的微扰展开重新表示为gN（'t Hooft）耦合的幂级数，我们给出了一些证据表明，在任意给定阶数n下，大N时随机过程分布存在趋于高斯性的倾向。通过对我们的级数求和，我们可以验证大N极限下能量及其（场论）方差的主阶结果。最后，我们建立了给定观测量（场论）方差的展开中各个微扰阶数与给定阶数NSPT随机过程的方差和协方差组合之间的一般关系。在确立了所有这些之后，我们最后讨论了NSPT计算在与O(N)类似的理论（即CP(N-1)模型）背景下的有趣应用。
英文摘要：
In the last three decades, Numerical Stochastic Perturbation Theory (NSPT) has proven to be an excellent tool for calculating perturbative expansions in theories such as Lattice QCD, for which standard, diagrammatic perturbation theory is known to be cumbersome. Despite the significant success of this stochastic method and the improvements made in recent years, NSPT apparently cannot be successfully implemented in low-dimensional models due to the emergence of huge statistical fluctuations: as the perturbative order gets higher, the signal to noise ratio is simply not good enough. This does not come as a surprise, but on very general grounds, one would expect that the larger the number of degrees of freedom, the less severe the fluctuations will be. By simulating $2D$ $O(N)$ non-linear sigma models for different values of $N$, we show that indeed the fluctuations are tamed in the large $N$ limit, meeting our expectations: for a large number of internal degrees of freedom (i.e. for large enough $N$), NSPT perturbative computation can be pushed to large perturbative orders $n$. By re-expressing our perturbative expansions as power series in the $gN$ ('t Hooft) coupling, we show some evidence that at any given order $n$ there is a tendency to gaussianity for the stochastic process distributions at large $N$. By summing our series, we can verify leading order results for the energy and its (field theoretic) variance in the large $N$ limit. We finally establish general relationships between the various perturbative orders in the expansion of the (field theoretic) variance of a given observable and combinations of variances and covariances of given orders NSPT stochastic processes. Having established all this, we conclude discussing interesting applications of NSPT computations in the context of theories similar to $O(N)$ (i.e. $CP(N-1)$ models).
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