Abstract
Last passage percolation and directed polymer models on are invariant under translation and certain reflections. When these models have an integrable structure coming from either the RSK correspondence or the geometric RSK correspondence (e.g., geometric last passage percolation or the log-gamma polymer), we show that these basic invariances can be combined with a decoupling property to yield a rich new set of symmetries. Among other results, we prove shift and rearrangement invariance statements for last passage times, geodesic locations, disjointness probabilities, polymer partition functions and quenched polymer measures. We also use our framework to find “scrambled” versions of the classical RSK correspondence and to find an RSK correspondence for moon polyominoes. The results extend to limiting models, including the KPZ equation and the Airy sheet.
Funding Statement
D. Dauvergne was supported by an NSERC PDF.
Acknowledgments
I would like to thank the organizers of the Banff International Research Station workshop on “Dimers, Ising Model and Interactions,” where the striking results of [7] were first brought to my attention. I would like to thank Ivan Corwin, Pavel Galashin, Vadim Gorin, Bálint Virág and Nikolaos Zygouras for helpful comments, discussions and for pointing out connections with other work. I would like to thank Sam Hopkins for suggesting that there might be a connection between this work and [24] and Hugh Thomas for helping me to understand this connection precisely.
Citation
Duncan Dauvergne. "Hidden invariance of last passage percolation and directed polymers." Ann. Probab. 50 (1) 18 - 60, January 2022. https://doi.org/10.1214/21-AOP1527
Information
