论文标题
严格的上限用于离散的bak-sneppen模型
Rigorous upper bound for the discrete Bak-Sneppen model
论文作者
论文摘要
修复[0,1] $中的一些$ p \和一个正整数$ n $。离散的bak-sneppen模型是Markov链，在带有周期性边界条件的长度$ n $零序列的空间上。在每一刻，最小元素（通常为零）以同样的概率选择，然后由独立的Bernoulli（P）随机变量与两个邻居一起替换。令$ν^{（n）}（p）$是该序列的元素等于此Markov链的固定分布下的元素的可能性。在[Barbay，Kenyon（2001）]中显示了$ν^{（n）}（p）\ to 1 $ as $ n \ to $ n \ to \ infty $时，$ p> 0.54 \ dots $;可惜的是，有证据不严格。 $ \ limsup的$ p \ in（0，p'）$ p'in（0，p'）$ for某些$ p'> 0 $的免费事实更难；最终在[Meester，Znamenski（2002）]中显示。 本说明的目的是通过证明$ν^{（n）}（p）\至1 $在$ p> 0.45 $时，提供了Barbay等人的严格证明以及改进。实际上，我们使用一些更精细的调整的方法也可以显示这一事实，即使所有$ p> 0.419533 $。
Fix some $p\in[0,1]$ 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 [Barbay, Kenyon (2001)] that $ν^{(n)}(p)\to 1$ as $n\to\infty$ when $p>0.54\dots$; the proof there is, alas, not rigorous. The complimentary fact that $\limsup ν^{(n)}(p)< 1$ for $p\in(0,p')$ for some $p'>0$ is much harder; this was eventually shown in [Meester, Znamenski (2002)]. The purpose of this note is to provide a rigorous proof of the result from Barbay et al, as well as to improve it, by showing that $ν^{(n)}(p)\to 1$ when $p>0.45$. In fact, our method with some finer tuning allows to show this fact even for all $p>0.419533$.
