# Large $N$ Limit of the $O(N)$ Linear Sigma Model via Stochastic Quantization

@article{Shen2020LargeL, title={Large \$N\$ Limit of the \$O(N)\$ Linear Sigma Model via Stochastic Quantization}, author={Hao Shen and Scott A. Smith and R. Zhu and Xiangchan Zhu}, journal={arXiv: Probability}, year={2020} }

This article studies large $N$ limits of a coupled system of $N$ interacting $\Phi^4$ equations posed over $\mathbb{T}^{d}$ for $d=1,2$, 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…

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