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 Φ4 equations posed over Td for d=2, known as the O(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/ N with respect to the Wasserstein distance. We also consider fluctuations and obtain tightness results for certain O(N) invariant observables, along with an exact description of the limiting correlations.