  In the zero temperature Brownian semi-discrete directed polymer we study the
joint distribution of two last-passage times at positions ordered in the
time-like direction. This is the situation when we have the slow de-correlation
phenomenon. We compute the limiting joint distribution function in a scaling
limit. This limiting distribution is given by an expansion in determinants
which is not a Fredholm expansion. A somewhat similar looking formula was
derived non-rigorously in a related model by Dotsenko.
