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Free boundary q-Whittaker and Hall-Littlewood processes

Jimmy He, Michael Wheeler

TL;DR

<3-5 sentence high-level summary>

Abstract

We study the free boundary $q$-Whittaker and Hall--Littlewood processes, two probability measures on sequences of partitions. We prove that a certain observable of the free boundary $q$-Whittaker process exhibits a $(q,t)$ symmetry after a random shift, generalizing a previous result of Imamura, Mucciconi, and Sasamoto, and an extension of that result due to the first author. Our proof is completely different, and as part of our proof, we find contour integral formulas for the free boundary $q$-Whittaker process. We also show a matching between certain observables in the free boundary Hall--Littlewood process and a quasi-open six vertex model, and explain how work of Finn and Vanicat gives an evaluation of a bounded sum over skew Hall--Littlewood functions as a rectangular Koornwinder polynomial.

Free boundary q-Whittaker and Hall-Littlewood processes

TL;DR

<3-5 sentence high-level summary>

Abstract

We study the free boundary -Whittaker and Hall--Littlewood processes, two probability measures on sequences of partitions. We prove that a certain observable of the free boundary -Whittaker process exhibits a symmetry after a random shift, generalizing a previous result of Imamura, Mucciconi, and Sasamoto, and an extension of that result due to the first author. Our proof is completely different, and as part of our proof, we find contour integral formulas for the free boundary -Whittaker process. We also show a matching between certain observables in the free boundary Hall--Littlewood process and a quasi-open six vertex model, and explain how work of Finn and Vanicat gives an evaluation of a bounded sum over skew Hall--Littlewood functions as a rectangular Koornwinder polynomial.

Paper Structure

This paper contains 35 sections, 25 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Rogers--Szegő polynomials
  5. The function $h_{\lambda}$
  6. Results on $q$-Whittaker processes
  7. Results on Hall--Littlewood processes
  8. Further directions
  9. Organization
  10. Preliminaries
  11. Macdonald polynomials
  12. Free boundary $q$-Whittaker and Hall--Littlewood processes
  13. Additional symmetries
  14. Equivalence of parameters and variables
  15. Type $D$ symmetry of the variables
  16. ...and 20 more sections

Key Result

Theorem 1.1

Fix an integer $n \geqslant 0$. The quantity is separately symmetric in the parameters $(q,t)$ and $(a,b,c,d)$.

Figures (1)

  • Figure 1: Example of part of a configuration of the quasi-open six vertex model. Note that arrows enter/exit from the diagonal boundaries.

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • ...and 38 more