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

Matsumoto, Sho

J. Math. Soc. Japan 57, No. 3, 619-637 (2005).

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.

MSC:

60C05	Combinatorial probability
05E05	Symmetric functions and generalizations

Keywords:

ascent pair; Tracy-Widom distribution; Schur \(Q\)-functions; Plancherel measure; limit distribution; random permutation; Hall-Littlewood functions

Citations:

Zbl 1060.60025