On a Local Version of the Bak–Sneppen Model

Ben-Ari, Iddo; Silva, Roger W. C.

doi:10.1007/s10955-018-2143-5

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…Definition 1 We say that the indices l and r flip when the configuration changes from ξ(t) to ξ(t + 1) such that the last event in (4) holds 5 .…”

Section: Remarkmentioning

confidence: 99%

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Rigorous Upper Bound for the Discrete Bak–Sneppen Model

Volkov

2021

J Stat Phys

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Fix some $$p\in [0,1]$$ p ∈ [ 0 , 1 ] and a positive integer n. The discrete Bak–Sneppen model is a Markov chain on the space of zero-one sequences of length n with periodic boundary conditions. At each moment of time a minimum element (typically, zero) is chosen with equal probability, and it is then replaced alongside both its neighbours by independent Bernoulli(p) random variables. Let $$\nu ^{(n)}(p)$$ ν ( n ) ( p ) be the probability that an element of this sequence equals one under the stationary distribution of this Markov chain. It was shown in Barbay and Kenyon (in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), pp. 928–933, SIAM, Philadelphia, PA, 2001) that $$\nu ^{(n)}(p)\rightarrow 1$$ ν ( n ) ( p ) → 1 as $$n\rightarrow \infty $$ n → ∞ when $$p>0.54\dots $$ p > 0.54 ⋯ ; the proof there is, alas, not rigorous. The complimentary fact that $$\displaystyle \limsup _{n\rightarrow \infty } \nu ^{(n)}(p)< 1$$ lim sup n → ∞ ν ( n ) ( p ) < 1 for $$p\in (0,p')$$ p ∈ ( 0 , p ′ ) for some $$p'>0$$ p ′ > 0 is much harder; this was eventually shown in Meester and Znamenski (J Stat Phys 109:987–1004, 2002). The purpose of this note is to provide a rigorous proof of the result from Barbay and Kenyon (in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), pp. 928–933, SIAM, Philadelphia, PA, 2001), as well as to improve it, by showing that $$\nu ^{(n)}(p)\rightarrow 1$$ ν ( n ) ( p ) → 1 when $$p>0.45$$ p > 0.45 . (Our method, in fact, shows that with some finer tuning the same is true for $$p>0.419533$$ p > 0.419533 .)

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“…Definition 1 We say that the indices l and r flip when the configuration changes from ξ(t) to ξ(t + 1) such that the last event in (4) holds 5 .…”

Section: Remarkmentioning

confidence: 99%

“…For some other recent results on the Bak-Sneppen model please see [3,5,7,8,10] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Rigorous Upper Bound for the Discrete Bak–Sneppen Model

Volkov

2021

J Stat Phys

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“…Example: ξ(t) = (1, 0, 0, 1, 0, 0), then Z t is either(1,2,3,4,5) or (4, 5, 0, 1, 2) 3. Example: n = 7, ξ = (1, 0, 0, 0, 0, 0, 1), (l, r) = (1, 5), D = 4.…”

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confidence: 99%

Rigorous upper bound for the discrete Bak-Sneppen model

Волков¹

2020

Preprint

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Fix some p ∈ [0, 1] and a positive integer n. The discrete Bak-Sneppen model is a Markov chain on the space of zero-one sequences of length n with periodic boundary conditions. At each moment of time a minimum element (typically, zero) is chosen with equal probability, and is then replaced together with both its neighbours by independent Bernoulli(p) random variables. Let ν (n) (p) be the probability that an element of this sequence equals one under the stationary distribution of this Markov chain. It was shown in [4] that ν (n) (p) → 1 as n → ∞ when p > 0.54 . . . ; the proof there is, alas, not rigorous. The complimentary fact that lim sup n→∞ ν (n) (p) < 1 for p ∈ (0, p ′ ) for some p ′ > 0 is much harder; this was eventually shown in [8].The purpose of this note is to provide a rigorous proof of the result from [4], as well as to improve it, by showing that ν (n) (p) → 1 when p > 0.45. (In fact, our method with some finer tuning allows to show this fact even for all p > 0.419533.

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A Local Barycentric Version of the Bak–Sneppen Model

Kennerberg

Volkov

2021

J Stat Phys

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We study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let $$N\ge 3$$ N ≥ 3 vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution $$\zeta $$ ζ . We show that in case where $$\zeta $$ ζ is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value.

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On a Local Version of the Bak–Sneppen Model

Cited by 4 publications

References 17 publications

Rigorous Upper Bound for the Discrete Bak–Sneppen Model

Rigorous Upper Bound for the Discrete Bak–Sneppen Model

Rigorous upper bound for the discrete Bak-Sneppen model

A Local Barycentric Version of the Bak–Sneppen Model

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