Phase Transitions in the Dynamics of Slow Random Monads
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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].

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 \((1p) \sqrt{n} + 5\).

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

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.
Keywords
Random monads Phase transition Random mappings Random dynamical systems Incomplete gamma function Recurrent TransientPreview
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