Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process

Abstract

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt (j ) be the number of particles which have crossed the bond from j to j +1 during the time span [0, t].For j = (1 - 2ρ)t+2ω(ρ(1 - ρ))1/3 t2/3 we prove that the fluctuations of Nt (j) for large t are of order t1/3 and we determine the limiting distribution function Fw(s), which is a generalization of the GUE Tracy-Widom distribution. The family Fw(s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In our work we arrive at Fw (s) through the asymptotics of a Fredholm determinant. Fw(s) is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle.