The Interaction Light Cone of the Discrete Bak–Sneppen, Contact and other local processes

Bannink, Tom; Buhrman, Harry; Gilyén, András; Szegedy, Márió

doi:10.1007/s10955-019-02351-y

Cited by 2 publications

(5 citation statements)

References 26 publications

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“…We say that the indices l and r flip when the configuration changes from ξ(t) to ξ(t + 1), and neither of equalities (2) holds. 3 .…”

Section: Resultsmentioning

confidence: 99%

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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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“…We say that the indices l and r flip when the configuration changes from ξ(t) to ξ(t + 1), and neither of equalities (2) holds. 3 .…”

Section: Resultsmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

Rigorous upper bound for the discrete Bak-Sneppen model

Волков¹

2020

Preprint

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“…If there is more than one such subset at time 0, choose any of them arbitrarily 3 . Now we can define…”

Section: Remarkmentioning

confidence: 99%

“…Hence, for every p > p we have T ( p, β ) < −ε( p) < 0. Finally, the cases where 3 ] < 0 on E t in the first case x = (0, 0, . .…”

Section: Proof Of Lemmamentioning

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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The Interaction Light Cone of the Discrete Bak–Sneppen, Contact and other local processes

Cited by 2 publications

References 26 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

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