Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall

We present an exact solution for the height distribution of KPZ equation at any time $t$ in a half space with flat initial condition. This is equivalent to obtaining free-energy polymer length pinned wall single point. In large limit binding transition takes place upon increasing attractiveness wall. Around critical point we find same statistics as Baik-Ben--Arous-P\'ech\'e outlier eigenvalues random matrix theory. bound phase, obtain measure endpoint and midpoint time. also unveil curious identities between partition functions half-space certain full Brownian-type