Uniform two-sided expansion sufficient for global synchrony

For any $M\in \mathbb{R}$, [F4] gives an $x>0$ with $L(x)>M$. For every positive integer $n\ge x$, monotonicity and identification give $\log n=L(n)\ge L(x)>M$. Thus $\log n\to \mathrm{\infty }$. Since $d\ge {\gamma }_0\log n$ and ${\gamma }_0>1$ is fixed, $d\to \mathrm{\infty }$ uniformly over the admissible $p$. Eventually $n\ge 3$, $d\ge \log n>0$ and $\alpha =d^{-1\mathrm{/}4}\le \min \{a\mathrm{/}2,1\}$, so both estimates in [F1] apply to [F3]. Outside the union of their failure events, all degrees are in $[ad,bd]$ and $\mathrm{\parallel }{\mathrm{\Delta }}_A\mathrm{\parallel }<d^{3\mathrm{/}4}=\alpha d$.

Write $D$ for the degree matrix. The definition yields the exact identity ${\mathrm{\Delta }}_L=L-dI+pJ=D-dI-{\mathrm{\Delta }}_A$. The diagonal bounds and [F2] therefore imply $(a-1-\alpha )dI⪯{\mathrm{\Delta }}_L⪯(b-1+\alpha )dI\mathrm{.}$ By step 1.1 these imply $(a\mathrm{/}2-1)dI⪯{\mathrm{\Delta }}_L⪯bdI$, as required. This uses the actual loop-free degrees and centering, including the $-pI$ shift in ${\mathrm{\Delta }}_A$.

Combining the adjacency and Laplacian conditions gives the declared expander event. The union bound in [F1] supplies its probability lower bound. The constants depend only on the fixed lower density ratio and not on the chosen sequence $p(n)$.