Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 9 / 3 / ii

The function $x{e}^{-x}$ is strictly increasing on $(0,1)$ and strictly decreasing on $(1,\infty )$. Hence for each $\lambda >1$ there is a unique $\theta ={\lambda }^{\ast }\in (0,1)$ with $\theta e^{-\theta }=\lambda e^{-\lambda }$.

Apply the subcritical component estimates to $G(n,\theta /n)$. For a specified vertex, the exact tree-component expectation in the Erdős-Rényi model gives the limiting probability that it lies in a tree component of order $j$:For each fixed $j$, the probability of a non-tree graph component of that size is $O_j(1/n)$: a spanning tree and one extra edge are necessary. Moreover the Galton-Watson process exploration estimate from the preceding proof bounds the probability of order greater than $K$ by $e^{-{\delta }_{\theta }(K-1)}$, uniformly in $n$, where ${\delta }_{\theta }=\theta -1-\log \theta >0$. First let $n\to \infty$ for fixed $K$, then let $K\to \infty$. No probability mass escapes to large components or non-tree components, so ${\sum }_{j\ge 1}{a}_j=1$. Multiplying by $\theta$ and using its defining identity givesEquivalently, the rooted-tree generating function selects the smaller solution of $T{e}^{-T}=\lambda e^{-\lambda }$. The choice of that branch is essential; the larger solution $\lambda$ is not the value of this convergent series.