Current fluctuations and large deviations for periodic TASEP on the relaxation scale

The one-dimensional totally asymmetric simple exclusion process (TASEP) with $N$ particles on a periodic lattice of $L$ sites is an interacting particle system with hopping rates breaking detailed balance. The total time-integrated current of particles $Q$ between time $0$ and time $T$ is studied for this model in the thermodynamic limit $L,N\to\infty$ with finite density of particles $\overline{\rho}=N/L$. The current $Q$ takes at leading order a deterministic value which follows from the hydrodynamic evolution of the macroscopic density profile by the inviscid Burgers' equation. Using asymptotics of Bethe ansatz formulas for eigenvalues and eigenvectors, an exact expression for the probability distribution of the fluctuations of $Q$ is derived on the relaxation time scale $T\sim L^{3/2}$ for an evolution conditioned on simple initial and final states. For flat initial and final states, a large deviation function expressed simply in terms of the Airy function is obtained at small rescaled time $T/L^{3/2}$.