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Annales de l’Institut Henri Poincaré D

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Volume 4, Issue 1, 2017, pp. 1–123
DOI: 10.4171/AIHPD/36

Published online: 2016-12-22

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

Konstantin Matveev[1] and Leonid Petrov[2]

(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).

For $0 Our new two-dimensional discrete time dynamics generalize and extend several known constructions. (1) The discrete time $q$-TASEPs studied by Borodin–Corwin [7] 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 [23]. (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 [59] and Borodin–Petrov [15], 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 [70] or strict-weak (introduced independently by O’Connell–Ortmann [58] and Corwin–Seppäläinen–Shen [25] 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