= Sharp subcritical largest-component scale
{title2=$L_1\sim\log n/(\lambda-1-\log\lambda),\quad0<\lambda<1$}
For fixed $0<\lambda<1$ in $G(n,\lambda/n)$, the <largest component of a graph> has the displayed order <with high probability>. A dominating <Galton-Watson process> gives the tail bound $\mathbb P(T\geq j)\leq e^{-(\lambda-1-\log\lambda)(j-1)}$ by the exponential <Markov inequality>. For the lower bound, the <tree-component expectation in the Erdős-Rényi model> diverges at $j=\lfloor(1-\eta)\log n/(\lambda-1-\log\lambda)\rfloor$. The disjoint-set covariance factor and the exclusion of overlaps give relative <variance> tending to zero, so the <second moment method> supplies a <tree component> of that order.
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