Tribes threshold window ($p_{1-\epsilon }-p_{\epsilon }={\Theta }_{\epsilon }(1/\log n)$)

For $m$ tribes of size $w$ with $m\sim (\log 2)2^w$ and $n=mw$, the success probability under the p-biased product measure is $1-(1-p^w{)}^m$. Its $x$-quantile is $p_x=[1-(1-x{)}^{1/m}{]}^{1/w}$. For fixed $0<x<1$,Thus the fixed-error window has order $1/\log n$. For small fixed error the leading coefficient grows with $\log (1/\epsilon )$, so the general Friedgut-Kalai sharp threshold theorem also has the correct error dependence up to absolute constants.