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

The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall