Duke Mathematical Journal

A limit theorem for shifted Schur measures

Craig A. Tracy and Harold Widom

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To each partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ 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 $m,n \to \infty$ with $\tau=m/n$ fixed. For the Schur measure, the $\alpha$-specialization limit law was derived by Johansson [J1]. Our main result implies that the two limit laws are identical.

Article information

Duke Math. J. Volume 123, Number 1 (2004), 171-208.

First available in Project Euclid: 13 May 2004

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 05E05: Symmetric functions and generalizations 05E10: Combinatorial aspects of representation theory [See also 20C30] 33E17: Painlevé-type functions 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]


Tracy, Craig A.; Widom, Harold. A limit theorem for shifted Schur measures. Duke Math. J. 123 (2004), no. 1, 171--208. doi:10.1215/S0012-7094-04-12316-4. http://projecteuclid.org/euclid.dmj/1084479322.

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