Phase Transitions in the Dynamics of Slow Random Monads

Abstract

Let us have a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B. Following Vladimir Arnold, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter in [0,1].

We fix a point ⊙∈B and consider the 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 depending on parameters n and s. In this study we define the term “mode” as we need. These are our main results showing existence of two phases in the following cases:

• On the one hand, if $s \ge1 / \sqrt{n}$, the mode of Vis equals 1 (which is the minimal value of Vis).

• On the other hand, if $s = p / \sqrt{n}$, where 0≤p<1, the mode of Vis is between $(1-\nobreak p) \sqrt{n} - 5$ and $(1-p) \sqrt{n} + 5$.

• On the one hand, if s≥1/2, the expectations of Vis and Rec are less than 2.

On the other hand:

• If $s \le1/\sqrt{n}$, the expectation of Vis is between $0.01 \cdot\sqrt{n}$ and $3 \cdot\sqrt{n}$ for all n>1.

• If s≤1/n, the expectation of Rec is between $\sqrt{\pi n/8} - 2$ and $\sqrt{\pi n/8} + 2$ for all n>1.