Height fluctuations for the stationary KPZ equation

Understand

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.