Annales de l’Institut Henri Poincaré D
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$q$-randomized Robinson–Schensted–Knuth correspondences and random polymersKonstantin Matveev and Leonid Petrov (1) Harvard University, Cambridge, USA
(2) Russian Academy of Sciences, Moscow, Russian Federation
We introduce and study $q$-randomized Robinson–Schensted–Knuth (RSK) correspondences which interpolate between the classical ($q=0$) and geometric $q \nearrow1$) RSK correspondences (the latter ones are sometimes also called tropical).
Our new two-dimensional discrete time dynamics generalize and extend several known constructions. (1) The discrete time $q$-TASEPs studied by Borodin–Corwin  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 . (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  and Borodin–Petrov , respectively. (3) In a scaling limit as $q\nearrow1$, two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences
associated with log-Gamma (introduced by Seppäläinen  or strict-weak (introduced independently by O’Connell–Ortmann  and Corwin–Seppäläinen–Shen  directed random lattice polymers.
Keywords: Robinson–Schensted–Knuth correspondence, random polymers,$q$-TASEP, Macdonald processes, random partitions, randomized insertion algorithm, interlacing particle arrays.
Matveev Konstantin, Petrov Leonid: $q$-randomized Robinson–Schensted–Knuth correspondences and random polymers. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 4 (2017), 1-123. doi: 10.4171/AIHPD/36