A Limit Theorem for Shifted Schur Measures

Abstract

To each partition $\lambda$ with distinct parts we assign the probability $Q_{\lambda }(x)P_{\lambda }(y)/Z$ where $Q_{\lambda }$ and $P_{\lambda }$ are the Schur $Q$-functions and $Z$ is a normalization constant. This measure, which we call the shifted Schur measure, is analogous to the much-studied Schur measure. For the specialization of the first $m$ coordinates of $x$ and the first $n$ coordinates of $y$ equal to $\alpha$ ($0<\alpha <1$) and the rest equal to zero, we derive a limit law for ${\lambda }_1$ as $m,n\ra\infty$ with $\tau =m/n$ fixed. For the Schur measure the $\alpha$-specialization limit law was derived by Johansson. Our main result implies that the two limit laws are identical.

35 pages, 2 figures. Version 3 adds a section on the Poisson limit of the shifted Schur measure

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