Phase Transitions in the Dynamics of Slow Random Monads

Ramos, A. D.; Toom, André

doi:10.1007/s10955-011-0373-x

Cited by 1 publication

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“…A similar statement was proved in [9], namely, if s ≥ 1/2, then E(Vis n ) ≤ 2. However, the present result is more general and precise.…”

Section: Lemma 4 For Any Fixed S and K The Quantity E(vis K N ) Is A supporting

confidence: 73%

“…It was proved in [9] that E(Rec n ) is a strictly decreasing function of s . Thus f (1, s) is a strictly decreasing function of s ∈ [0, 1].…”

Section: Lemmamentioning

confidence: 99%

“…It was proved in [9] that the mode and expectations of Vis n , Rec n and Tra n undergo a kind of phase transition. Our present task is to study the moments of Vis n , Rec n and Tra n depending on parameters n > 1 and s ∈ [0, 1].…”

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confidence: 99%

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Moments and Distributions of Trajectories in Slow Random Monads

Ramos

Toom

2012

J Stat Phys

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Given a finite set B (basin) with n > 1 elements, which we call points, and a map M : B → B, 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. Here s is a parameter, 0 ≤ s ≤ 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 n , Rec n and Tra n the numbers of visited, recurrent and transient points respectively. We prove that, when n tends to infinity, Vis n and Tra n converge in law to geometric distributions and Rec n converges in law to a distribution concentrated at its lowest value, which is one. Now about moments. The case s = 1 is trivial, so let 0 ≤ s < 1. For any natural number k there is a number such that the k-th moments of Vis n , Rec n and Tra n do not exceed this number for all n. About Vis n : for any natural k the k-th moment of Vis n is an increasing function of n. So it has a limit when n → ∞ and for all n it is less than this limit. About Rec n : for any k the k-th moment of Rec n tends to one when n tends to infinity. About Tra n : for any k the k-th moment of Tra n has a limit when n tends to infinity.

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“…A similar statement was proved in [9], namely, if s ≥ 1/2, then E(Vis n ) ≤ 2. However, the present result is more general and precise.…”

Section: Lemma 4 For Any Fixed S and K The Quantity E(vis K N ) Is A supporting

confidence: 73%

“…It was proved in [9] that E(Rec n ) is a strictly decreasing function of s . Thus f (1, s) is a strictly decreasing function of s ∈ [0, 1].…”

Section: Lemmamentioning

confidence: 99%