The discrete periodic Pitman transform: invariances, braid relations, and Burke properties

TL;DR

This paper develops the theory of the discrete periodic Pitman transform, demonstrating its braid relations, group actions, and invariance properties in polymer models, extending previous results to periodic and full-line cases.

Contribution

It introduces the discrete periodic Pitman transform, proves its braid relations and invariance properties, and applies these to polymer models with periodic environments.

Findings

Transform satisfies braid relations similar to full-line case

Polymer partition functions are invariant under the transform

Multi-path invariance extends to full-line limit

Abstract

We develop the theory of the discrete periodic Pitman transform, first introduced by Corwin, Gu, and the fifth author. We prove that the discrete periodic Pitman transform satisfies the same braid relations that are satisfied for the full-line Pitman transform shown by Biane, Bougerol, and O'Connell. This defines a group action of the infinite symmetric group on sequences of vectors in $\mathbb R^{\mathbb Z_N}$. We prove that, for polymers in a periodic environment, single-path and multi-path partition functions are preserved under the action of this transform on the weights in the polymer model. Combined with a new inhomogeneous Burke property for the periodic Pitman transform, we prove a multi-path invariance result for the periodic inverse-gamma polymer under permutations of the column parameters. In the limit to the full-line case, we obtain a multi-path extension of a recent invariance result of Bates, Emrah, Martin, Sepp\"al\"ainen, and the fifth author, in both positive and zero-temperature.