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 Φ 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/ √ 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 [SSZZ20] from two dimensions to three dimensions. The proof leverages the method recently developed by [GH18] and combines many new techniques such as uniform in N estimates on perturbative objects as well as the solutions.

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