\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Sho Matsumoto}
%
%
\medskip
\noindent
%
%
{\bf Jack Deformations of Plancherel Measures and Traceless Gaussian Random Matrices}
%
%
\vskip 5mm
\noindent
%
%
%
%
We study random partitions
$\lambda=(\lambda_1,\lambda_2,\dots,\lambda_d)$ of $n$ whose length is
not bigger than a fixed number $d$.  Suppose a random partition
$\lambda$ is distributed according to the Jack measure, which is a
deformation of the Plancherel measure with a positive parameter
$\alpha>0$.  We prove that for all $\alpha>0$, in the limit as $n \to
\infty$, the joint distribution of scaled $\lambda_1,\dots, \lambda_d$
converges to the joint distribution of some random variables from a
traceless Gaussian $\beta$-ensemble with $\beta=2/\alpha$.  We also
give a short proof of Regev's asymptotic theorem for the sum of
$\beta$-powers of $f^\lambda$, the number of standard tableaux of
shape $\lambda$.



\bye

.
