Random Mappings with Ewens Cycle Structure

Abstract

In this paper, we consider a random mapping \(\hat{T}_{n,\theta}\) of the finite set \(\{1,2,\ldots,n\}\) into itself, for which the digraph representation \(\hat{G}_{n,\theta}\) is constructed by: (1) selecting a random number \(\hat{L}_n\) of cyclic vertices, (2) constructing a uniform random forest of size \(n\) with the selected cyclic vertices as roots, and (3) forming `cycles’ of trees by applying to the selected cyclic vertices a random permutation with cycle structure given by the Ewens sampling formula with parameter \(\theta\). We investigate \(\hat{k}_{n,\theta}\), the size of a `typical’ component of \(\hat{G}_{n,\theta}\), and we obtain the asymptotic distribution of \(\hat{k}_{n,\theta}\) conditioned on \(\hat{L}_n = m(n)\). As an application of our results, we show in Section 3 that provided \(\hat{L}_n\) is of order much larger than \(\sqrt{n}\), then the joint distribution of the normalized order statistics of the component sizes of \(G_{n,\theta}\) converges to the Poisson-Dirichlet \((\theta)\) distribution as \(n \to \infty\).