  We present new probabilistic and combinatorial identities relating three
random processes: the oriented swap process on $n$ particles, the corner growth
process, and the last passage percolation model. We prove one of the
probabilistic identities, relating a random vector of last passage percolation
times to its dual, using the duality between the Robinson-Schensted-Knuth and
Burge correspondences. A second probabilistic identity, relating those two
vectors to a vector of 'last swap times' in the oriented swap process, is
conjectural. We give a computer-assisted proof of this identity for $n\le 6$
after first reformulating it as a purely combinatorial identity, and discuss
its relation to the Edelman-Greene correspondence. The conjectural identity
provides precise finite-$n$ and asymptotic predictions on the distribution of
the absorbing time of the oriented swap process, thus conditionally solving an
open problem posed by Angel, Holroyd and Romik.
