  We prove Airy process variational formulas for the one-point probability
distribution of (discrete time parallel update) TASEP with general initial
data, as well as last passage percolation from a general lattice path to a
point. We also consider variants of last passage percolation with inhomogeneous
parameter geometric weights and provide variational formulas of a similar
nature. This proves one aspect of the conjectural description of the
renormalization fixed point of the Kardar-Parisi-Zhang universality class.
