Let (Xn) be a residual allocation model with i.i.d. residual fractions Un: For W a random variable with values in [0; 1] and independent of (Xn), we define another sequence (Yn) by setting

(formula here)

Under minor regularity assumptions we show that (Xn) and (Yn) have the same probability law if and only if this law is a GEM distribution. In this case, the distribution of W and the Uns is Beta(1; θ) for some θ > 0.

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.