Half-Space Stationary Kardar-Parisi-Zhang Equation

We study the solution of the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height h(x, t) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at $$ . The boundary condition $$ corresponds to an attractive wall for $$ , and leads to the binding of the polymer to the wall below the critical value $$ . Here we choose the initial condition h(x, 0) to be a Brownian motion in $$ with drift $$ . When $$ , the solution is stationary, i.e. $$ remains at all times a Brownian motion with the same drift, up to a global height shift h(0, t). We show that the distribution of this height shift is invariant under the exchange of parameters A and B. For any $$ , we provide an exact formula characterizing the distribution of h(0, t) at any time t, using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters A, B. In particular, when $$ , the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik-Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea-Ferrari-Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.