Séminaire Lotharingien de Combinatoire, 84B.3 (2020), 12 pp.

Elia Bisi, Fabio Deelan Cunden, Shane Gibbons and Dan Romik

Sorting Networks, Staircase Young Tableaux, and Last Passage Percolation

Abstract. We present new combinatorial and probabilistic 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 <= 6 after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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