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diff --git a/posts/2016-10-13-q-robinson-schensted-knuth-polymer.org b/posts/2016-10-13-q-robinson-schensted-knuth-polymer.org new file mode 100644 index 0000000..93da639 --- /dev/null +++ b/posts/2016-10-13-q-robinson-schensted-knuth-polymer.org @@ -0,0 +1,50 @@ +#+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:%7B%7B%20site.url%20%7D%7D/assets/resources/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. |