q-randomized Robinson-Schensted-Knuth correspondences and random polymers

TL;DR

This paper introduces q-randomized RSK correspondences that interpolate between classical and geometric RSK, linking combinatorics, probability, and integrable systems, and extends known models to new q-parameterized dynamics.

Contribution

The paper develops new q-randomized RSK correspondences and associated Markov dynamics that unify and extend classical and geometric RSK, connecting to integrable particle systems and polymers.

Findings

Derived four Markov dynamics generalizing classical RSK.

Connected dynamics to q-TASEP and q-PushTASEP particle systems.

Established a Fredholm determinant formula for q-PushASEP.

Abstract

We introduce and study q-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical (q=0) and geometric (q->1) RSK correspondences (the latter ones are sometimes also called tropical). For 0<q<1 our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a certain class of probability measures on arrays, namely, on q-Whittaker processes (which are t=0 versions of Macdonald processes). We present four Markov dynamics which for q=0 reduce to the classical row or column RSK correspondences applied to a random input matrix with independent geometric or Bernoulli entries. Our new two-dimensional discrete time dynamics generalize and extend several known constructions: (1) The discrete time q-TASEPs arise as one-dimensional marginals of our "column" dynamics. In a similar way, our "row" dynamics lead to discrete time q-PushTASEPs - new integrable particle systems in the Kardar-Parisi-Zhang universality class. We employ these new one-dimensional discrete time systems to establish a Fredholm determinantal formula for the two-sided continuous time q-PushASEP conjectured by Corwin-Petrov (2013). (2) In a certain Poisson-type limit (from discrete to continuous time), our two-dimensional dynamics reduce to the q-randomized column and row Robinson-Schensted correspondences introduced by O'Connell-Pei (2012) and Borodin-Petrov (2013), respectively. (3) In a scaling limit as q->1, two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma or strict-weak directed random polymers.