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We consider the totally asymmetric simple exclusion process (TASEP) withtwo-sided Bernoulli initial condition, i.e., with left density rho_- and rightdensity rho_+. We consider the associated height function, whose discretegradient is given by the particle occurrences. Macroscopically one has adeterministic limit shape with a shock or a rarefaction fan depending on thevalues of rho_{+/-}. We characterize the large time scaling limit of thefluctuations as a function of the densities rho_{+/-} and of the differentmacroscopic regions. Moreover, using a slow decorrelation phenomena, theresults are extended from fixed time to the whole space-time, except along thesome directions (the characteristic solutions of the related Burgers equation)where the problem is still open.  On the way to proving the results for TASEP, we obtain the limit processesfor the fluctuations in a class of corner growth processes with externalsources, of equivalently for the last passage time in a directed percolationmodel with two-sided boundary conditions. Additionally, we provide analogousresults for eigenvalues of perturbed complex Wishart (sample covariance)matrices.

Limit processes for TASEP with shocks and rarefaction fans