Up to a random height shift, two-sided Brownian motion is invariant for the Kardar-Parisi-Zhang equation. In this talk we describe recent results with Borodin, Ferrari and Veto through which we use Macdonald processes and the geometric Robinson-Schensted-Knuth correspondence to compute the distribution of this height shift and demonstrate cube-root fluctuations in large time, with a universal limit law. This also relates to the two-point correlation function and super-diffusivity of the stochastic Burgers equation.-