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We present a probabilistic generalization of the Robinson--Schenstedcorrespondence in which a permutation maps to several different pairs ofstandard Young tableaux with nonzero probability. The probabilities depend ontwo parameters $q$ and $t$, and the correspondence gives a new proof of thesquarefree part of the Cauchy identity for Macdonald polynomials (i.e., theequality of the coefficients of $x_1 \cdots x_n y_1 \cdots y_n$ on either side,which are related to permutations and standard Young tableaux). By specializing$q$ and $t$ in various ways, one recovers the row and column insertion versionsof the Robinson--Schensted correspondence, several $q$- and $t$-deformations ofrow and column insertion which have been introduced in recent years inconnection with $q$-Whittaker and Hall--Littlewood processes, and thePlancherel measure on partitions. Our construction is based on Fomin's growthdiagrams and the recently introduced notion of a probabilistic bijectionbetween weighted sets.

$q{\rm RS}t$: A probabilistic Robinson--Schensted correspondence for Macdonald polynomials