A SCALING LIMIT FOR t-SCHUR MEASURES

We introduce a new measure on partitions. We assign to each partition $ lambda $ a probability $ S_lambda (x;t)s_lambda (y)/Z_t $ where $s_lambda $ is the Schur function, $ S_lambda (x;t) $ is a generalization of the Schur function defined by Macdonald (Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford, 1995) and $ Z_t $ is a normalization constant. This measure, which we call the $ t $-Schur measure, is a generalization of the Schur measure (A. Okounkov. Infinite wedge and random partitions. Selecta Math. (N.S.) 7 (2001), 57-81) and is closely related to the shifted Schur measure studied by Tracy and Widom (A limit theorem for shifted Schur measures. Preprint (2002), math.PR/0210255) for a combinatorial viewpoint. We prove that a limit distribution of the length of the first row of a partition with respect to $ t $-Schur measures is given by the Tracy-Widom distribution, Le. the limit distribution of the largest eigenvalue suitably centered and normalized in the Gaussian unitary ensemble