This article studies large $N$ limits of a coupled system of $N$ interacting $\Phi^4$ 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 $1Λsqrt{N}$ 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. 

　　Publication: 

　　The Annals of Probability 2022, Vol. 50, No. 1, 131–202 

　　Author: 

　　Hao Shen 

　　Department of Mathematics, University of Wisconsin - Madison, USA 

　　E-mail: pkushenhao@gmail.com  

　　Scott Smith 

　　Department of Mathematics, University of Wisconsin - Madison, USA 

　　E-mail: ssmith74@wisc.edu  

　　Rongchan Zhu 

　　Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China; Fakult？t für Mathematik, Universit？t Bielefeld, D-33501 Bielefeld, Germany 

　　E-mail: zhurongchan@126.com  

　　Xiangchan Zhu 

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; Fakult？t für Mathematik, Universit？t Bielefeld, D-33501 Bielefeld, Germany 

　　E-mail: zhuxiangchan@amss.ac.cn