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One-ended spanning trees in amenable unimodular graphs. (English) Zbl 1457.60020
A unimodular random graph is a probability measure on a collection of locally finite graphs with a special vertex considered as their root, where the mass transport principle holds. That is, the expected mass that is sent out of the roots is equal to the total mass that is received by the root. A subgraph \(H\) of a rooted graph \((G,\ast)\) is called a factor of iid (abbreviated as fiid), if it can be constructed as a measurable function from iid uniformly distributed random labels of \(V(G)\) which belong to \([0,1]\). The main result of this paper is the almost sure existence of a factor of iid spanning trees with one end in a unimodular random graph that is amenable and ergodic and has one end. A characterisation of amenability in relation to fiids is that a unimodular random graph \(G\) is amenable if and only if it contains a factor of iid subgraphs \(\Gamma_n\), with finite components almost surely, whose union is \(G\). The main theorem gives also a characterisation of the amenability of a quasi-transitive unimodular amenable graph with one end as having an invariant spanning tree with one end.

60C05 Combinatorial probability
05C80 Random graphs (graph-theoretic aspects)
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