A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-polymer
Abstract
In [Matveev-Petrov 2016](arXiv:1504.00666) a -deformed Robinson-Schensted-Knuth algorithm (RSK) was introduced. In this article we give reformulations of this algorithm in terms of the Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pairs are swapped in a sense of distribution when the input matrix is transposed. We also formulate a -polymer model based on the RSK, 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 -geometric weights. We use the -local moves to define a generalisation of the 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 -polymer in -geometric environment, formulate a -version of the multilayer polynuclear growth model (PNG) and write down the joint distribution of the -polymer partition functions at a fixed time.
Cite
@article{arxiv.1610.03692,
title = {A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-polymer},
author = {Yuchen Pei},
journal= {arXiv preprint arXiv:1610.03692},
year = {2017}
}
Comments
Revised version after two rounds of review: added 5 figures; expanded Section 4.2 about qPNG; fixed many typos; clear up the expositions; and more