论文标题
通过随机量化的$ O（n）$线性Sigma模型的$ n $限制
Large $N$ Limit of the $O(N)$ Linear Sigma Model via Stochastic Quantization
论文作者
论文摘要
本文研究了$ n $互动$φ^4 $方程的耦合系统的大$ n $限制，该方程在$ \ mathbb {t}^{d} $ for $ d = 2 $上，称为$ O（n）$ linearear sigma模型。建立了动力学的$ n $边界的统一，使我们能够向平均场奇异SPDE展示融合，也被证明是全球范围的。此外，我们在$ n $限制中表现出不变措施的紧密度。对于足够大的质量，它们会收敛到（大型）高斯自由场，这是平均场动力学的独特量度，以$ 1/\ sqrt {n} $相对于Wasserstein距离而言。我们还考虑波动并获得某些$ o（n）$不变的可观察结果的紧密结果，以及对限制相关性的确切描述。
This article studies large $N$ limits of a coupled system of $N$ interacting $Φ^4$ equations posed over $\mathbb{T}^{d}$ for $d=2$, known as the $O(N)$ linear sigma model. Uniform in $N$ bounds on the dynamics are established, allowing us to show convergence to a mean-field singular SPDE, also proved to be globally well-posed. Moreover, we show tightness of the invariant measures in the large $N$ limit. For large enough mass, they converge to the (massive) Gaussian free field, the unique invariant measure of the mean-field dynamics, at a rate of order $1/\sqrt{N}$ with respect to the Wasserstein distance. We also consider fluctuations and obtain tightness results for certain $O(N)$ invariant observables, along with an exact description of the limiting correlations.
