Large N limit of the O(N) linear sigma model via stochastic quantization

Abstract

This article studies large N limits of a coupled system of N interacting ${\mathrm{\Phi }}^4$ equations posed over ${\mathbb{T}}^{\mathit{d}}$ for $\mathit{d}=2$, known as the $\mathit{O}(\mathit{N})$ 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 $1/\sqrt{\mathit{N}}$ with respect to the Wasserstein distance. We also consider fluctuations and obtain tightness results for certain $\mathit{O}(\mathit{N})$ invariant observables, along with an exact description of the limiting correlations.