Title: Large $N$ limit and $1/N$ expansion of invariant observables in $O(N)$ linear $σ$-model via SPDE Show Chinese title

Title: 大$N$极限和不变可观测量在$O(N)$线性$σ$-模型中的$1/N$展开通过 SPDE

Abstract: In this paper, we continue the study of large $N$ problems for the Wick renormalized linear sigma model, i.e. $N$-component $\Phi^4$ model, in two spatial dimensions, using stochastic quantization methods and Dyson--Schwinger equations. We identify the large $N$ limiting law of a collection of Wick renormalized $O(N)$ invariant observables. In particular, under a suitable scaling, the quadratic observables converge in the large $N$ limit to a mean-zero (singular) Gaussian field denoted by $\mathcal{Q}$ with an explicit covariance; and the observables which are renormalized powers of order $2n$ converge in the large $N$ limit to suitably renormalized $n$-th powers of $\mathcal{Q}$. The quartic interaction term of the model has no effect on the large $N$ limit of the field, but has nontrivial contributions to the limiting law of the observables, and the renormalization of the $n$-th powers of $\mathcal{Q}$ in the limit has an interesting finite shift from the standard one. Furthermore, we derive the $1/N$ asymtotic expansion for the $k$-point functions of the quadratic observables by employing graph representations and analyzing the order of each graph from Dyson--Schwinger equations. Finally, turning to the stationary solutions to the stochastic quantization equations, with the Ornstein--Uhlenbeck process being the large $N$ limiting dynamic, we derive here its next order correction in stationarity, as described by an SPDE with the right-hand side having explicit fixed-time marginal law which involves the above field $\mathcal{Q}$. Show Chinese abstract

Abstract: 在本文中，我们继续研究二维空间中Wick正规化的线性σ模型的大型$N$问题，即$N$组分$\Phi^4$模型，使用随机量化方法和Dyson--Schwinger方程。我们确定了一组Wick正规化的$O(N)$不变可观测量的大型$N$极限定律。 特别是，在适当的缩放下，二次可观测量在大$N$极限下收敛到一个均值为零（奇异）的高斯场，记为$\mathcal{Q}$，其协方差是显式的；而被归一化的阶为$2n$的可观测量在大$N$极限下收敛到适当归一化的$n$次幂$\mathcal{Q}$。 模型的四次相互作用项对场的大$N$极限没有影响，但对可观测量的极限分布有非平凡的贡献，在极限中$n$次幂的$\mathcal{Q}$的归一化有一个有趣的有限偏移，与标准情况不同。此外，我们通过采用图表示并分析来自 Dyson--Schwinger 方程的每个图的阶数，推导出二次可观测量的$k$点函数的$1/N$渐近展开式。最后，转向随机量化方程的平稳解，其中 Ornstein--Uhlenbeck 过程是大$N$极限动态，我们在此推导出其平稳状态下的下一个阶修正，该修正由一个具有显式固定时间边缘分布的 SPDE 描述，该分布涉及上述场$\mathcal{Q}$。