The totally asymmetric simple exclusion process (TASEP) is one of the
simplest models of interacting particle systems on the one-dimensional
lattice. It is equivalent to a random growth model from the
Kardar-Parisi-Zhang universality class. We focus on fluctuations of the
particle positions for a nonequilibrium TASEP that starts from certain
deterministic initial conditions. We (rigorously) derive the scaling
exponents 1/3 and 2/3, and identify the limit laws as those of Gaussian
Orthogonal and Unitary ensembles of the random matrix theory.