Abstract
We discover a new property of the stochastic colored six-vertex model called flip-invariance. We use it to show that for a given collection of observables of the model, any transformation that preserves the distribution of each individual observable also preserves their joint distribution. This generalizes recent shift-invariance results of Borodin–Gorin–Wheeler. As limiting cases, we obtain similar statements for the Brownian last passage percolation, the Kardar–Parisi–Zhang equation, the Airy sheet and directed polymers. Our proof relies on an equivalence between the stochastic colored six-vertex model and the Yang–Baxter basis of the Hecke algebra. We conclude by discussing the relationship of the model with Kazhdan–Lusztig polynomials and positroid varieties in the Grassmannian.
Funding Statement
This work was partially supported by an Alfred P. Sloan Research Fellowship and by the National Science Foundation under Grants DMS-1954121 and DMS-2046915.
Acknowledgments
I am deeply grateful to Alexei Borodin for sparking my interest in this problem and for his guidance throughout the various stages of the project. I am also indebted to Vadim Gorin for the numerous consultations and explanations. Additionally, I would like to thank Thomas Lam and Pavlo Pylyavskyy with whom I discussed some questions and objects from Section 7. Finally, I am grateful to the anonymous referees for their extremely careful reading of this manuscript and many suggested improvements.
Citation
Pavel Galashin. "Symmetries of stochastic colored vertex models." Ann. Probab. 49 (5) 2175 - 2219, September 2021. https://doi.org/10.1214/20-AOP1502
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