Hidden invariance of last passage percolation and directed polymers

Duncan Dauvergne

Published 2020-02-21Version 1

Last passage percolation and directed polymer models on $\mathbb{Z}^2$ are invariant under translation and certain reflections. When these models have an integrable structure (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, quenched polymer measures, and geodesic locations. We also use our framework to find new `scrambled' versions of the classical RSK correspondence, and find an RSK correspondence for moon polyominoes. The results extend to limiting models, including the KPZ equation and the Airy sheet.