Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Eva R. Engel, Benjamin Jasper Kra-Caskey, Oleksandr Lazorenko, Caio Hermano Maia de Oliveira, Evan Sorensen, Ivan Wong, Ryan Xu, Xinyi Zhang

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, Seppäläinen, and the fifth author, in both positive and zero-temperature.

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

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 . 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, Seppäläinen, and the fifth author, in both positive and zero-temperature.

Paper Structure

This paper contains 19 sections, 24 theorems, 90 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The positive-temperature discrete periodic Pitman transform
  3. Polymers on $\mathbb{Z}^2$ and periodic environments
  4. Invariance of periodic polymer partition functions
  5. Permutation invariance
  6. Zero-temperature limits
  7. Organization of the paper
  8. Notation and conventions
  9. Acknowledgements
  10. Algebraic Properties of the Pitman Transform
  11. Matrix encoding
  12. Proof of Theorem \ref{['thm:braid']}
  13. Proof of Item \ref{['it:involution']}:
  14. Proof of Item \ref{['it:braid']}
  15. Invariance of polymer partition functions
  16. ...and 4 more sections

Key Result

Theorem 1.1

The operators $\mathcal{P}_k$ satisfy the following:

Figures (5)

  • Figure 1: Representation of indices $[i,j]$, demonstrating periodicity.
  • Figure 2: Example multi-path in $\mathbb{Z}^2$ from $\{(0,0),(2,0)\}$ to $\{(2,4),(4,4)\}$
  • Figure 3: Two multipaths in $\mathbb{Z}^2$ from $U:=\{u_1,u_2\}$ (green) to $V:=\{v_1,v_2\}$ (orange), where $(U,V)\notin \Psi$. To see this, note that there exists a multipath from $(u_1,u_2)$ to $(v_1,v_2)$ (denoted here in blue/thick) as well as a multipath from $(u_1,u_2)$ to $(v_2,v_1)$ (denoted here in red/thin).
  • Figure 4: An example of an pair $(U,V) \in \mathcal{W}^\sigma$, where $U$ and $V$ each contain $3$ elements. Each column corresponds to a periodic vector of weights $\mathbf X_k$. The gray horizontal lines are used to indicate a periodic environment with $N = 4$. In this figure, the column parameters of columns $3,5$, and $6$ (blue) are permuted, and the column parameters in columns $8$ and $9$ (red) are permuted. The condition is that, for each of the three end points $(x,y)$ in $U$ (lower left), $\sigma(\mathbb{Z}_{<x}) = \mathbb{Z}_{<x}$, while for each of the three end points $(x,y)$ in $V$ (upper right), $\sigma(\mathbb{Z}_{>x}) = \mathbb{Z}_{>x}$. The multi-path partition function for this pair is preserved after changing the weights from $\mathbf X$ to $\sigma \mathbf X$, and the distributional equality in Theorem \ref{['thm:per_permutation-invariance']} holds jointly over all end point pairs $(U,V) \in \Psi$ satisfying this property.
  • Figure 5: Local transformation of a vertex-weighted node (left) with weight $X_{(r,s)}$ into an edge-weighted directed graph (right), where the internal edge carries weight $\omega_{(r,s)}=\exp(X_{(r,s)})$ and external edges carry weight 1. Paths in the original graph correspond to paths from "in" nodes to "out" nodes in the transformed graph, preserving path weights.

Theorems & Definitions (53)

  • Theorem 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 43 more