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 $1∧sqrt{N}$ 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.
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