Sorting networks, staircase Young tableaux and last passage percolation
Elia Bisi, Fabio Deelan Cunden, Shane Gibbons, Dan Romik

TL;DR
This paper establishes new combinatorial and probabilistic identities connecting the oriented swap process, corner growth process, and last passage percolation, using advanced combinatorial correspondences and computer-assisted proofs.
Contribution
It introduces novel identities linking three stochastic processes and proves one using duality and combinatorial methods, with conjectures supported by computational evidence.
Findings
Proved a probabilistic identity relating last passage percolation times and their duals.
Formulated a conjectural identity connecting swap times and last passage times.
Provided a computer-assisted proof for small system sizes.
Abstract
We present new combinatorial and probabilistic identities relating three random processes: the oriented swap process on 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 after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence.
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Taxonomy
TopicsRandom Matrices and Applications · Advanced Combinatorial Mathematics · Stochastic processes and statistical mechanics
