Random initial data and average shock time in the Fermi-Pasta-Ulam-Tsingou chain

We investigate the dynamics of the Fermi--Pasta--Ulam--Tsingou chain with long-wavelength random initial data. When the energy per particle is small, thermal equilibrium is not reached on a fast timescale and the system enters prethermalization. The formation of the prethermal state is characterized by the development of a Burgers-type shock and the onset of a turbulent-like spectrum with a time dependent exponent $\zeta(t)$ in the inertial range. We perform a significant step forward by demonstrating that these features are robust under generic long-wavelength random initial conditions. By employing advanced probabilistic techniques inspired by the works of Dudley and Talagrand, we derive a sharp asymptotic expression for the average shock time in the thermodynamic limit. For large $p$, this time scales as $(p \sqrt{\log p})^{-1}$, where $p$ is the number of excited modes proving that it is an intensive quantity up to a logarithmic correction in the size of the system.