Trajectories in Random Monads

Abstract

Let us have a non-empty finite set S with n>1 elements which we call points and a map M:S→S. After V.I. Arnold, we call such pairs (S, M) monads, but we consider random monads in which all the values of M(⋅) are random, independent and uniformly distributed in S. We fix some ⊙∈S and consider the infinite sequence M t (⊙), t=0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis, Rec, Tra the numbers of visited, recurrent and transient points respectively and study their distributions. The distributions of Vis, Rec, Tra are unimodal. The modes of Rec and Tra equal their minimal values, that is 1 and 0 respectively. The mode of Vis is approximated by $\sqrt{n}$ , plus-minus a constant. The mathematical expectations: $\mathbb{E}(\mathit{Vis})$ is approximated by $2 \sqrt{\pi\, n/8}$ plus-minus a constant; $\mathbb{E}(\mathit{Rec})$ and $\mathbb{E}(\mathit{Tra})$ are approximated by $\sqrt{\pi\, n/8}$ plus-minus a constant. For the standard deviations σ(Vis) and σ(Rec)=σ(Tra) respectively we present the approximations $$\sqrt{\frac{4-\pi}{2} \cdot n} \quad\mbox{and}\quad \sqrt{\frac{16-3\pi}{24} \cdot n},$$ from which they also deviate at most by a constant. We prove that when n tends to infinity, the correlations Corr(Rec,Tra) and Corr(Rec,Vis)=Corr(Tra,Vis) converge to $$\frac{8-3\pi}{16-3\pi}\quad \mbox{and}\quad \sqrt{\frac{12-3\pi}{16-3\pi}}.$$