Correlation functions of the shifted Schur measure. (English) Zbl 1078.60010

Summary: The shifted Schur measure introduced by C. A. Tracy and H. Widom [Duke Math. J. 123, 171–208 (2004; Zbl 1060.60025)] is a measure on the set of all strict partitions, which is defined by Schur \(Q\)-functions. The main aim of this paper is to calculate the correlation function of this measure, which is given by a Pfaffian. As an application, we prove that a limit distribution of parts of partitions with respect to a shifted version of the Plancherel measure for symmetric groups is identical with the corresponding distribution of the original Plancherel measure. In particular, we obtain a limit distribution of the length of the longest ascent pair for a random permutation. Further we give expressions of the mean value and the variance of the size of partitions with respect to the measure defined by Hall-Littlewood functions.


60C05 Combinatorial probability
05E05 Symmetric functions and generalizations


Zbl 1060.60025
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