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The oriented swap process is a natural directed random walk on the symmetricgroup that can be interpreted as a multi-species version of the TotallyAsymmetric Simple Exclusion Process (TASEP) on a finite interval. An openproblem from a 2008 paper of Angel, Holroyd, and Romik asks for the limitingdistribution of the absorbing time of the process. We resolve this question byproving that this random variable satisfies GOE Tracy-Widom asymptotics. Ourstarting point is a distributional identity relating the behavior of theoriented swap process to last passage percolation, conjectured in a recentpaper of Bisi, Cunden, Gibbons, and Romik. The main technical tool is ashift-invariance principle for multi-species TASEPs, obtained by exploitingrecent results of Borodin, Gorin, and Wheeler for the stochastic coloredsix-vertex model.

Absorbing time asymptotics in the oriented swap process