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, corner growth process, last passage percolation (LPP) model. prove one of identities, a vector LPP times to its dual, using duality between Robinson–Schensted–Knuth Burge correspondences. A second identity, those two vectors “last times” in OSP, is conjectural. give computer-assisted proof this identity for after first reformulating it as purely discuss relation Edelman–Greene correspondence. The conjectural provides precise finite-n asymptotic predictions distribution absorbing time thus conditionally solving an open problem posed by Angel, Holroyd, Romik.