The Plancherel measure of the Young-Fibonacci graph

Abstract

The Young-Fibonacci graph YF 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 off 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.