Absorbing time asymptotics in the oriented swap process

The oriented swap process is a natural directed random walk on the symmetric group that can be interpreted as a multispecies version of the totally asymmetric simple exclusion process (TASEP) on a finite interval. An open problem from a 2009 paper of Angel, Holroyd, and Romik asks for the limiting distribution of the absorbing time of the process as the number of particles goes to infinity. We resolve this question by proving that this random variable satisfies GOE Tracy–Widom asymptotics. As a central ingredient of our proof, we reexamine 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. We use a shift-invariance principle for multispecies TASEPs, obtained by exploiting recent results of Borodin, Gorin, and Wheeler for the stochastic colored six-vertex model, to prove a weakened form of the Bisi et al. conjectural identity, that is nonetheless sufficient for proving the asymptotic result for the absorbing time.