Large N Limit of the O(N) Linear Sigma Model in 3D

In this paper we study the large N limit of O(N)-invariant linear sigma model, which is a vector-valued generalization $$\Phi ^4$$ quantum field theory, on three dimensional torus. We problem via its stochastic quantization, yields coupled system interacting SPDEs. prove tightness invariant measures in limit. For enough mass or small coupling constant, they converge to (massive) Gaussian free at rate order $$1/\sqrt{N}$$ with respect Wasserstein distance. also obtain results for certain O(N) observables. These generalize some Shen et al. (Ann Probab 50(1):131–202, 2022) from two dimensions dimensions. The proof leverages method recently developed by Gubinelli and Hofmanová (Commun Math Phys 384(1):1–75, 2021) combines many new techniques such as uniform estimates perturbative objects well solutions.