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

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 01, two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma or strict-weak directed random polymers.