The oriented swap process and last passage percolation

We present new probabilistic and combinatorial identities relating three random processes: the oriented swap process (OSP) on n particles, the corner growth process, and the last passage percolation (LPP) model. We prove one of the probabilistic identities, relating a random vector of LPP 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 OSP, is conjectural. We give a computer-assisted proof of this identity for n <= 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 OSP, thus conditionally solving an open problem posed by Angel, Holroyd, and Romik.