The oriented swap process (OSP) is a model for a random sorting network, with \$N\$ particles labeled \$1, \dots, N\$ performing successive adjacent swaps at random times until they reach the reverse configuration \$N,\dots, 1\$. We prove a conjectured distributional equality between the OSP and the exponential Last Passage Percolation model. Using this, we also get new asymptotic results about the OSP, some of which are in connection with the Airy sheet.
The distributional identity can be viewed as a special case of a new shift-invariance of the colored TASEP. It is related to recent results on symmetries of the six-vertex models (by Borodin-Gorin-Wheeler and Galashin). Our proof uses analyticity of the transition probability functions and induction arguments.