Synchronization for sufficiently large logarithmic density constants

PDF p. 31: Paragraph after equation (6.2)

Assume countable choice. For every fixed $0<\epsilon <1$, there is ${\gamma }_0={\gamma }_0(\epsilon )>1$ such that, uniformly for $p\in [{\gamma }_0\log n\mathrm{/}n,1]$, with probability tending to one as $n\to \mathrm{\infty }$, the simple graph $G(n,p)$ is an $(n,d^{\ast },\alpha ,c^{-},c^{+})$-expander with $d^{\ast }=np,\alpha =(2+\epsilon )(np{)}^{-1\mathrm{/}2},c^{-}=-\epsilon ,c^{+}=\epsilon ,$ and satisfies the hypotheses of Theorem 1.10; hence it is globally synchronizing. Uniformity means the supremum of the failure probability over the stated $p$ interval tends to zero; it includes every varying sequence $p=p(n)$ in that range.

The original large-density claim on PDF p. 31, after (6.2), is retained: for sufficiently large ${\gamma }_0$, $\gamma \ge {\gamma }_0$ gives $d^{\ast }=pn$, $\alpha =(2+\epsilon )(pn{)}^{-1\mathrm{/}2}$, $c^{-}=-\epsilon$, $c^{+}=\epsilon$, the Theorem 1.10 condition and global synchrony with high probability for $p\ge \gamma \log n\mathrm{/}n$. The explicit range $0<\epsilon <1$ spells out its fixed small-parameter convention. The local proof preserves its rich parameter conclusion and makes the probability uniformity precise. The local proof uses Uniform constant-two adjacency control above a large logarithmic density and Uniform degree slack and the vanishing synchrony condition. The article and the fixed-law Péché–Soshnikov result are not logical suppliers for the spectral estimate.