Shift-invariance for vertex models and polymers

Abstract

We establish a symmetry in a variety of integrable stochastic systems: Certain multi-point distributions of natural observables are unchanged under a shift of a subset of observation points. The property holds for stochastic vertex models, (1+1)d directed polymers in random media, last passage percolation, the Kardar-Parisi-Zhang equation, and the Airy sheet. In each instance it leads to computations of previously inaccessible joint distributions. The proofs rely on a combination of the Yang-Baxter integrability of the inhomogeneous colored stochastic six-vertex model and Lagrange interpolation. We also show that a simplified (Gaussian) version of our theorems is related to the invariance in law of the local time of the Brownian bridge under the shift of the observation level.

arXiv comment102 pages. v2: misprints corrected

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• Open ASEP in the Weakly Asymmetric Regime
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Cited by in corpus (10)

• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Limiting current distribution for a two species asymmetric exclusion process
• Observables of stochastic colored vertex models and local relation
• Transition probability and total crossing events in the multi-species asymmetric exclusion process
• Hidden diagonal integrability of $q$-Hahn vertex model and Beta polymer model
• Parameter Permutation Symmetry in Particle Systems and Random Polymers
• The oriented swap process and last passage percolation
• Inhomogeneous spin $q$-Whittaker polynomials
• Shift-Invariance of the Colored TASEP and Finishing Times of the Oriented Swap Process
• Hidden symmetries of weighted lozenge tilings