Large $N$ limit of the $O(N)$ linear sigma model in 3D

In this paper we study the large N limit of the $O(N)$-invariant linear sigma model, which is a vector-valued generalization of the $\Phi^4$ quantum field theory, on the three dimensional torus. We study the problem via its stochastic quantization, which yields a coupled system of N interacting SPDEs. We prove tightness of the invariant measures in the large N limit. For large enough mass or small enough coupling constant, they converge to the (massive) Gaussian free field at a rate of order $1/\sqrt N$ with respect to the Wasserstein distance. We also obtain tightness results for certain $O(N)$ invariant observables. These generalize some of the results in \cite{SSZZ20} from two dimensions to three dimensions. The proof leverages the method recently developed by \cite{GH18} and combines many new techniques such as uniform in $N$ estimates on perturbative objects as well as the solutions.