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#+title: A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer

#+date: <2016-10-13>

(Latest update: 2017-01-12) In
[[http://arxiv.org/abs/1504.00666][Matveev-Petrov 2016]] a
\(q\)-deformed Robinson-Schensted-Knuth algorithm (\(q\)RSK) was
introduced. In this article we give reformulations of this algorithm in
terms of Noumi-Yamada description, growth diagrams and local moves. We
show that the algorithm is symmetric, namely the output tableaux pair
are swapped in a sense of distribution when the input matrix is
transposed. We also formulate a \(q\)-polymer model based on the
\(q\)RSK and prove the corresponding Burke property, which we use to
show a strong law of large numbers for the partition function given
stationary boundary conditions and \(q\)-geometric weights. We use the
\(q\)-local moves to define a generalisation of the \(q\)RSK taking a
Young diagram-shape of array as the input. We write down the joint
distribution of partition functions in the space-like direction of the
\(q\)-polymer in \(q\)-geometric environment, formulate a \(q\)-version
of the multilayer polynuclear growth model (\(q\)PNG) and write down the
joint distribution of the \(q\)-polymer partition functions at a fixed
time.

This article is available at
[[https://arxiv.org/abs/1610.03692][arXiv]]. It seems to me that one
difference between arXiv and Github is that on arXiv each preprint has a
few versions only. In Github many projects have a "dev" branch hosting
continuous updates, whereas the master branch is where the stable
releases live.

[[file:/assets/qrsklatest.pdf][Here]]
is a "dev" version of the article, which I shall push to arXiv when it
stablises. Below is the changelog.

- 2017-01-12: Typos and grammar, arXiv v2.
- 2016-12-20: Added remarks on the geometric \(q\)-pushTASEP. Added
  remarks on the converse of the Burke property. Added natural language
  description of the \(q\)RSK. Fixed typos.
- 2016-11-13: Fixed some typos in the proof of Theorem 3.
- 2016-11-07: Fixed some typos. The \(q\)-Burke property is now stated
  in a more symmetric way, so is the law of large numbers Theorem 2.
- 2016-10-20: Fixed a few typos. Updated some references. Added a
  reference: [[http://web.mit.edu/~shopkins/docs/rsk.pdf][a set of notes
  titled "RSK via local transformations"]]. It is written by
  [[http://web.mit.edu/~shopkins/][Sam Hopkins]] in 2014 as an
  expository article based on MIT combinatorics preseminar presentations
  of Alex Postnikov. It contains some idea (applying local moves to a
  general Young-diagram shaped array in the order that matches any
  growth sequence of the underlying Young diagram) which I thought I was
  the first one to write down.