3 The sample and its law
A field of offspring counts is a map \(c\colon \mathcal{N}(N)\to \mathbb {N}\). The sample of \(c\) is the subtree in which each vertex \(v\) keeps its first \(c(v)\) children,
The field survives when \(\mathcal{T}(c)\) is infinite.
The residual subtree of \(\mathcal{T}(c)\) at a vertex of the sample is the sample of the shifted field, and \(\mathcal{T}(c)\) decomposes into its root together with the samples hanging off the children of the root.
For words \(v\) and \(w\), the inequalities defining membership of \(vw\) split at \(\lvert v \rvert \). The inequalities before the split say that \(v\in \mathcal{T}(c)\), while those after the split say that \(w\) belongs to the sample of the shifted field \(c_v(u)=c(vu)\). Thus
When \(v\in \mathcal{T}(c)\), this equivalence identifies the residual subtree at \(v\) with \(\mathcal{T}(c_v)\). Taking \(v\) to be a one-letter word and separating the empty word gives the decomposition at the root.
The skeleton of a sample is the set of its vertices whose residual subtree is infinite.
A skeleton vertex has a child in the skeleton, and hence carries a ray, an isometric copy of \(\mathbb {N}\) inside the sample.
Let \(v\) lie in the skeleton. By lemma 3.2, its infinite residual subtree is the union of its root and the residual subtrees below its finitely many children. At least one of these child subtrees is therefore infinite, and the corresponding child of \(v\) again lies in the skeleton. Iterating this choice gives vertices \(r(n)\) with \(r(0)=v\) and \(r(n+1)\) a child of \(r(n)\).
If \(m\leq n\), then \(r(m)\) is a prefix of \(r(n)\) and their lengths differ by \(n-m\). The distance formula in lemma 1.3 consequently gives \(d(r(m),r(n))=n-m\). Symmetry yields \(d(r(m),r(n))=\lvert m-n \rvert \) for all \(m,n\), so \(r\) is an isometric ray contained in the skeleton.
The field measure on \(\mathcal{N}(N)\to \mathbb {N}\) is the infinite product of copies of \(\theta \), one per vertex. Its pushforward along \(\mathcal{T}\) is the law of the Galton–Watson tree of \(\theta \), a measure on subtrees for the \(\sigma \)-algebra generated by the vertex events \(\{ T : v\in T\} \).
If \(J\leq N\), then \(\mathbb {P}(\mathcal{T}\text{ finite})=q\).
Let \(E_n\) be the event that the sample has no vertex in generation \(n\). Conditioning on the root having \(j\) children and using the independence of the \(j\) shifted fields gives
Summing over \(j\) yields \(\mathbb {P}(E_{n+1})=f(\mathbb {P}(E_n))\). Since \(E_0\) is empty, induction gives \(\mathbb {P}(E_n)=f^n(0)\).
The sequence \(f^n(0)\) is increasing and bounded by every fixed point of \(f\) in \([0,1]\). Its limit is a fixed point by continuity, and is therefore the least fixed point \(q\). The events \(E_n\) are increasing, and their union is the event that the finitely branching sample is finite. Continuity of measure from below now gives \(\mathbb {P}(\mathcal{T}\text{ finite})=q\).