Height Fluctuations for the Stationary KPZ Equation

We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data H(0, X) = B(X), for B(X) a two-sided standard Brownian motion) and show that as time T goes to infinity, the fluctuations of the height function H(T, X) grow like T-1/3 and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula for Whittaker processes. We relate this to a polymer model which mixes the semi-discrete and log-gamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a two-sided Brownian motion with drift beta to the left of the origin and b to the right of the origin. The Fredholm determinant has a limit for beta > b, and the case where beta = b ( corresponding to the stationary initial data) follows from an analytic continuation argument.