Random Mappings with a Given Number of Cyclical Points

Abstract

In this paper, we consider a random mapping, \(\hat{T}_n\), of the finite set \(\{1,2,\ldots,n\}\) into itself for which the digraph representation \(\hat{G}_n\) 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 a random permutation to the selected cyclic vertices.We investigate \(\hat{k}_n\), the size of a `typical’ component of \(\hat{G}_n\), and, under the assumption that the random permutation on the cyclical vertices is uniform, we obtain the asymptotic distribution of \(k\), 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 \(\hat{G}_n\) converges to the Poisson-Dirichlet \((1)\) distribution as \(n \to \infty\). Other applications and generalizations are also discussed in Section \(3\).