  This article studies several properties of the half-space last passage
percolation, in particular the two-time covariance. We show that, when the two
end-points are at small macroscopic distance, then the first order correction
to the covariance for the point-to-point model is the same as the one of the
stationary model. In order to obtain the result, we first derive comparison
inequalities of the last passage increments for different models. This is used
to prove tightness of the point-to-point process as well as localization of the
geodesics. Unlike for the full-space case, for half-space we have to overcome
the difficulty that the point-to-point model in half-space with generic start
and end points is not known.
