Rigorous upper bound for the discrete Bak-Sneppen model

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 $\nu^{(n)}(p)$ 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 $\nu^{(n)}(p)\to 1$ as $n\to\infty$ when $p>0.54\dots$; the proof there is, alas, not rigorous. The complimentary fact that $\limsup \nu^{(n)}(p)< 1$ 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 $\nu^{(n)}(p)\to 1$ when $p>0.45$. In fact, our method with some finer tuning allows to show this fact even for all $p>0.419533$.