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

Abstract

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 Shen et al. (Ann Probab 50(1):131–202, 2022) from two dimensions to three dimensions. The proof leverages the method recently developed by Gubinelli and Hofmanová (Commun Math Phys 384(1):1–75, 2021) and combines many new techniques such as uniform in N estimates on perturbative objects as well as the solutions.