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| author | Yuchen Pei <me@ypei.me> | 2018-04-06 17:43:24 +0200 |
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| committer | Yuchen Pei <me@ypei.me> | 2018-04-06 17:43:24 +0200 |
| commit | 2a2c61de0e44adad26c0034dfda6594c34f0d834 (patch) | |
| tree | 75d8d3960b552cf3b8b56e0abf11e78ca28f8776 /site/posts/2016-10-13-q-robinson-schensted-knuth-polymer.html | |
| parent | 76ab6c66b3c65f16c8d19a6d16c100cf45ec9e57 (diff) | |
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diff --git a/site/posts/2016-10-13-q-robinson-schensted-knuth-polymer.html b/site/posts/2016-10-13-q-robinson-schensted-knuth-polymer.html new file mode 100644 index 0000000..93bd833 --- /dev/null +++ b/site/posts/2016-10-13-q-robinson-schensted-knuth-polymer.html @@ -0,0 +1,37 @@ +<!doctype html> +<html lang="en"> + <head> + <meta charset="utf-8"> + <title>A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer</title> + <link rel="stylesheet" href="../assets/css/default.css" /> + <script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> + </head> + <body> + <header> + <span class="logo"> + <a href="../blog.html">Blog</a> + </span> + <nav> + <a href="../index.html">About</a><a href="../postlist.html">All posts</a><a href="../blog-feed.xml">Feed</a> + </nav> + </header> + + <div class="main"> + <div class="bodyitem"> + <h2> A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer </h2> + <p>Posted on 2016-10-13</p> + <p>(Latest update: 2017-01-12) In <a href="http://arxiv.org/abs/1504.00666">Matveev-Petrov 2016</a> 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.</p> +<p>This article is available at <a href="https://arxiv.org/abs/1610.03692">arXiv</a>. 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.</p> +<p><a href="%7B%7B%20site.url%20%7D%7D/assets/resources/qrsklatest.pdf">Here</a> is a “dev” version of the article, which I shall push to arXiv when it stablises. Below is the changelog.</p> +<ul> +<li>2017-01-12: Typos and grammar, arXiv v2.</li> +<li>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.</li> +<li>2016-11-13: Fixed some typos in the proof of Theorem 3.</li> +<li>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.</li> +<li>2016-10-20: Fixed a few typos. Updated some references. Added a reference: <a href="http://web.mit.edu/~shopkins/docs/rsk.pdf">a set of notes titled “RSK via local transformations”</a>. It is written by <a href="http://web.mit.edu/~shopkins/">Sam Hopkins</a> 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.</li> +</ul> + + </div> + </div> + </body> +</html> |