  Exciton-polaritons under driven-dissipative conditions exhibit a condensation
transition which belongs to a different universality class than equilibrium
Bose-Einstein condensates. By numerically solving the generalized
Gross-Pitaevskii equation with realistic experimental parameters, we show that
one-dimensional exciton-polaritons display fine features of Kardar-Parisi-Zhang
(KPZ) dynamics. Beyond the scaling exponents, we show that their phase
distribution follows the Tracy-Widom form predicted for KPZ growing interfaces.
We moreover evidence a crossover to the stationary Baik-Rains statistics. We
finally show that these features are unaffected on a certain timescale by the
presence of a smooth disorder often present in experimental setups.
