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

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