Hyperbolic $O(N)$ linear sigma model and its mean-field limit

We study large $N$ limits of the hyperbolic $O(N)$ linear sigma model (${\text{HLSM}}_N$) on the two-dimensional torus ${\mathbb{T}}^2$, namely, a system of $N$ interacting stochastic damped nonlinear wave equations (SdNLW) with coupled cubic nonlinearities. After establishing (pathwise) global well-posedness of ${\text{HLSM}}_N$ and the limiting equation, called the mean-field SdNLW, we first establish global-in-time convergence of ${\text{HLSM}}_N$ to the mean-field SdNLW with general initial data (under a suitable assumption). In particular, for the local-in-time convergence, we obtain an optimal convergence rate of order $N^{-\frac{1}{2}}$ under an additional integrability assumption on initial data. We then show that the invariant Gibbs dynamics for ${\text{HLSM}}_N$ converges to that for the mean-field SdNLW with a convergence rate of order $N^{-\frac{1}{2}}$ on any large time intervals.