In this talk, I will discuss large \(N\) limits of a coupled system of \(N\) interacting \(\Phi^4\) equations posed over \(\mathbb{T}^{d}\) for \(d=1,2,3\),
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, I 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 \(1/\sqrt{N}\) with respect to
the Wasserstein distance.

I will also consider fluctuations and obtain tightness results for certain \(O(N)\) invariant observables, along with an exact description of the limiting
correlations in \(d=1,2\).

This talk is based on joint work with Hao Shen, Scott Smith and Xiangchan Zhu.