Erdős–Rényi global synchronization — Theorem 1.2

Set ${\gamma }_0=1+\epsilon$ and $d=np$. The uniform expansion result [F3] supplies fixed $a,b,\eta >0$, $c^{-}=a\mathrm{/}2-1\in (-1,0)$ and $c^{+}=b>0$, with an expander event of probability at least $1-K{d}^{-1\mathrm{/}4}-n^{-\eta }-n^{-2}$. For every $M\in \mathbb{R}$, [F4] supplies $x>0$ with $L(x)>M$; every positive integer $n\ge x$ then satisfies $\log n=L(n)\ge L(x)>M$. Hence $\log n\to \mathrm{\infty }$, so $d\ge {\gamma }_0\log n\to \mathrm{\infty }$ and $d>0$ eventually. Thus the expander event has probability tending to one along this arbitrary sequence.

Apply the numerical-limit lemma in [F3] to these fixed $c^{\pm }$. Eventually $0<\alpha =d^{-1\mathrm{/}4}\le 1\mathrm{/}5$ and the spectral criterion's numerical expression is strictly below one. All hypotheses of [F2], including positive reference degree, now hold on the expander event.

By [F2], every graph in that event is globally synchronizing under the assumed countable choice. The synchronization probability is therefore at least the probability in step 1.1, which tends to one. This proves the complete fixed-$\epsilon$ statement, including varying density and $p=1$.