Journal of Statistical Physics

, Volume 145, Issue 5, pp 1324–1342 | Cite as

Phase Transitions in the Dynamics of Slow Random Monads

  • A. D. Ramos
  • A. Toom


Let us have a finite set B (basin) with n>1 elements, which we call points, and a map M:BB. 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,bB the probability of M(a)=a is s and the probability of M(a)=b, where ab, 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.


Random monads Phase transition Random mappings Random dynamical systems Incomplete gamma function Recurrent Transient 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972) MATHGoogle Scholar
  2. 2.
    Adell, J.A., Jodrá, P.: The median of Poisson distribution. Metrika 6, 337–346 (2005) CrossRefGoogle Scholar
  3. 3.
    Arnold, V.I.: Topology of algebra: combinatorics of the operation of squaring. Funct. Anal. Appl. 37(3), 20–35 (2003) (In Russian) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Arnold, V.I.: Complexity of finite sequences of zeros and ones and geometry of finite functional spaces. Public lecture delivered on May 13 (2006) (In Russian). Available at
  5. 5.
    Arnold, V.I.: Topology and statistics of arithmetic and algebraic formulae. Preprint available at
  6. 6.
    Jaworski, J.: Epidemic processes on digraphs of random mappings. J. Appl. Probab. 36(3), 780–798 (1999) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kac, L.: Probability of indecomposability of a random mapping function. Ann. Math. Stat. 26, 512–517 (1955) CrossRefGoogle Scholar
  8. 8.
    Kruskal, M.D.: The expected number of components under a random mapping function. Am. Math. Mon. 61, 392–397 (1954) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Metropolis, N., Ulam, S.: A Property of randomness of an arithmetical function. Am. Math. Mon. 60, 252–253 (1953) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Ramos, A., Toom, A.: Trajectories in random monads. J. Stat. Phys. 142, 201–219 (2011) CrossRefMATHADSMathSciNetGoogle Scholar
  11. 11.
    Stirling’s approximation, Wikipedia Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of StatisticsFederal University of PernambucoRecifeBrazil

Personalised recommendations