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Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. (English) Zbl 1118.82032
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 J. Baik and 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.

##### MSC:
 82C22 Interacting particle systems in time-dependent statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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