A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-polymer

TL;DR

This paper extends the $q$-RSK algorithm, introduces a $q$-polymer model, and explores their properties, including symmetry, Burke property, and law of large numbers, with applications to $q$-geometric weights and growth models.

Contribution

It provides reformulations, symmetry proofs, and a new $q$-polymer model based on the $q$-RSK algorithm, along with distributional results and generalizations.

Findings

The $q$-RSK algorithm is symmetric under transposition.

A $q$-polymer model with Burke property is formulated.

A strong law of large numbers for the $q$-polymer partition function is proved.

Abstract

In [Matveev-Petrov 2016](arXiv:1504.00666) a $q$-deformed Robinson-Schensted-Knuth algorithm ($q$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 $q$-polymer model based on the $q$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 $q$-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.