Hidden invariance of last passage percolation and directed polymers

TL;DR

This paper uncovers new symmetries in last passage percolation and directed polymers with integrable structures, extending classical invariances to more complex models and their limits.

Contribution

It introduces a framework combining invariance and decoupling to reveal new symmetries and correspondences in integrable probabilistic models.

Findings

Proved shift and rearrangement invariance for last passage times and polymer measures.

Discovered scrambled RSK correspondences and RSK for moon polyominoes.

Extended invariance results to KPZ equation and Airy sheet.

Abstract

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 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.