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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 26/1998

Alexander GNEDIN and Sergei KEROV
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THE PLANCHEREL MEASURE
OF THE YOUNG-FIBONACCI GRAPH

This preprint was accepted November, 1998

Contact:
` Alexander GNEDIN and Sergei KEROV `

ABSTRACT:
The Young-Fibonacci graph $\Bbb{YF}$ is an important example
(along with the Young lattice) of differential posets studied by
S.~Fomin and R.~Stanley. For every differential poset there is a
distinguished central measure called the Plancherel measure. We
study the Plancherel measure and the associated Markov chain, the
Plancherel process, on the Young-Fibonacci graph.
We establish a law of large numbers which implies that the
Plancherel measure cannot be represented as a nontrivial mixture
of central measures, i.e. is ergodic. Our second result claims
the convergence of the level distributions of the Plancherel
measure to the GEM(1/2) probability law in the space of
nonnegative series with unit sum, which is a particular example
of distribution from the class of Residual Allocation Models.
In order to obtain the Plancherel process as an image of a
sequence of independent uniformly distributed random variables,
we establish a new version of the Robinson-Schensted type
correspondence between permutations and pairs of paths in the
Young-Fibonacci graph. This correspondence is used to demonstrate
a recurrence property of the Plancherel process.

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