= Uniform spanning tree of a recurrent infinite graph
= Infinite recurrent uniform spanning tree
{synonym}
On a connected locally finite <recurrent graph>, root <Wilson's algorithm> at a fixed <graph vertex> $r$ and process a countable enumeration of the <graph vertices>. Every walk hits the existing <tree> <almost surely>, since it contains $r$. Each attached <graph path> is finite, and the union is a connected acyclic spanning subgraph. Its law is the infinite <uniform spanning tree>.
It is also the common free and wired finite-volume <limit of a sequence> along every <graph exhaustion>. For a finite <edge> set, run a finite initial segment of the chosen enumeration containing all its endpoints. Its membership is then permanently decided, because later attachments have new internal <graph vertices>. Only finitely many finite random-walk trajectories were used, and their <graph vertices> and <graph neighbours> fit inside all sufficiently large exhaustion sets. Coupling then gives convergence of those finite-dimensional <edge> laws for both boundary conventions. Finite <Wilson's algorithm> is root/order independent, so the limiting law is too. There is no uniform counting measure on all infinite <spanning trees>; the definition is this limiting <probability> law.
Back to article page
