Abstract
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.
Funding Statement
A.B. was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—GZ 2047/1, Projekt ID 390685813.
V.G. was partially supported by NSF Grants DMS-1664619, DMS-1949820, by the NEC Corporation Fund for Research in Computers and Communications, and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin–Madison with funding from the Wisconsin Alumni Research Foundation.
D.R. was supported by NSF Grant DMS-1800725.
Acknowledgements
The authors thank the Institute for Pure and Applied Mathematics (IPAM) at UCLA for its hospitality during their visit there in February 2020, where some of the ideas contained in the current work were discussed. We also thank Leonid Petrov for helpful discussions, and the anonymous referees for helpful suggestions.
V. Gorin is also affiliated with the Institute for Information Transmission Problems of Russian Academy of Sciences.
Citation
Alexey Bufetov. Vadim Gorin. Dan Romik. "Absorbing time asymptotics in the oriented swap process." Ann. Appl. Probab. 32 (2) 753 - 763, April 2022. https://doi.org/10.1214/21-AAP1695
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