Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process (Q863130)

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli \(\rho\) measure as initial conditions, \(0< \rho <1\), is stationary in space and time. Let \(N_t(j)\) be the number of particles which have crossed the bond from \(j\) to \(j+1\) during the time span \([0,t]\). For \( j = (1-2p)t + 2w(\rho(1-\rho))^{1/3}t^{2/3}\) the authors prove that the fluctuations of \(N_t(j)\) for large \(t\) are of order \(t^{1/3}\) and determine the limiting distribution function \(F_w(s)\), which is a generalization of the GUE (Gaussian unitary ensemble) Tracy-Widom distribution. The family \(F_w(s)\) of distribution functions have been obtained before by \textit{J. Baik} and \textit{E. M. Rains} [J. Stat. Phys. 100, 523--541 (2000; Zbl 0976.82043)] in the context of the PNG model (polynuclear growth model) with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In this work the authors arrive at \(F_w(s)\) through the asymptotics of a Fredholm determinant. \(F_w(s)\) 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.