The oriented swap process and last passage percolation

Abstract

AbstractWe 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 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.