Shift-invariance for vertex models and polymers

TL;DR

This paper uncovers a shift-invariance symmetry across various integrable stochastic systems, enabling new computations of joint distributions and linking to Brownian bridge local time invariance.

Contribution

It introduces a novel shift-invariance property for multiple models, leveraging Yang-Baxter integrability and interpolation techniques, expanding understanding of stochastic vertex models and polymers.

Findings

Shift-invariance holds for multiple stochastic models.

Enables computation of previously inaccessible joint distributions.

Links to invariance of Brownian bridge local time.

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.