Bisi, E., Cunden, F. D., Gibbons, S., & Romik, D. (2022). The oriented swap process and last passage percolation. Random Structures and Algorithms, 60(4), 690–715. https://doi.org/10.1002/rsa.21055
Applied Mathematics; Software; General Mathematics; Computer Graphics and Computer-Aided Design
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Abstract:
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 urn:x-wiley:rsa:media:rsa21055:rsa21055-math-0001 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.
