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arXiv · 2003.03331

Sorting networks, staircase Young tableaux and last passage percolation

Elia Bisi·Fabio Deelan Cunden·Shane Gibbons·Dan Romik
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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 $n\le 6$ after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence.

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Elia Bisi, Fabio Deelan Cunden, Shane Gibbons, Dan Romik. 2020-06-05. Sorting networks, staircase Young tableaux and last passage percolation. https://arxiv.org/abs/2003.03331

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