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

Understand

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).