On Real Time Dynamics of Large $N$ Models

We analyze the real-time dynamics of the large $N$ vector model, focusing on heavy states with energies of the order $N$. In this regime, we demonstrate that interactions become sufficiently strong to produce non-zero condensate of the Hubbard-Stratonovich field $\sigma$, which, in turn, induces particle production. This process leads to a significant transformation of the initial state and potential thermalization. For homogeneous perturbations, our results show that the equations become integrable, yet can still lead to thermalization in the continuum limit. Furthermore, we calculate the energies of these heavy states and their contributions to the thermal free energy, thereby determining the free energy of the critical $O(N)$ model by operator counting.