Sorting networks, staircase Young tableaux and last passage percolation

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 $n$ 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.