  The oriented swap process is a natural directed random walk on the symmetric
group that can be interpreted as a multi-species version of the Totally
Asymmetric Simple Exclusion Process (TASEP) on a finite interval. An open
problem from a 2008 paper of Angel, Holroyd, and Romik asks for the limiting
distribution of the absorbing time of the process. We resolve this question by
proving that this random variable satisfies GOE Tracy-Widom asymptotics. Our
starting point is a distributional identity relating the behavior of the
oriented swap process to last passage percolation, conjectured in a recent
paper of Bisi, Cunden, Gibbons, and Romik. The main technical tool is a
shift-invariance principle for multi-species TASEPs, obtained by exploiting
recent results of Borodin, Gorin, and Wheeler for the stochastic colored
six-vertex model.
