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The Plancherel measure of the Young–Fibonacci graph

Published online by Cambridge University Press:  17 January 2001

ALEXANDER GNEDIN  and
SERGEI KEROV
ALEXANDER GNEDIN
Affiliation:
Institute of Mathematical Stochastics, University of Göttingen, Lotzestrasse 13, 37083 Göttingen, Germany; e-mail: gnedin@math.uni-goettingen.de
SERGEI KEROV
Affiliation:
Steklov Mathematical Institute at St. Petersburg (POMI), Fontanka 27, St. Petersburg, 191011 Russia; e-mail: kerov@pdmi.ras.ru
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Abstract

The Young–Fibonacci graph [ ][ ] is an important example (along with the Young lattice) of differential posets studied by Fomin and 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(½) 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 3 , November 2000 , pp. 433 - 446
Copyright
© 2000 Cambridge Philosophical Society

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