Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 214 / 4 / a

Use the usual implicit convention that the infinite connected graph is a locally finite graph. This is needed for the simple random walk and the finite wired constructions to be defined. Choose finite vertex sets $V_1\subset V_2\subset \cdots$ exhausting $V$, with each induced subgraph $G[V_k]$ connected.

For the free uniform spanning forest, sample a uniform spanning tree of $G[V_k]$, using only the internal edges. Its limiting law on edge subsets of $G$ is the free uniform spanning forest measure:The limit is weak convergence of probability measures on $\{0,1{\}}^E$, equivalently convergence of every finite-edge event.

For the wired uniform spanning forest, form $G_k^{\mathrm{w}}$ by identifying every vertex outside $V_k$ to one boundary vertex ${\partial }_k$. Retain all edges with at least one endpoint in $V_k$, preserve parallel edges, and discard loops at ${\partial }_k$. Sample a uniform spanning tree of this finite multigraph and restrict it to the original internal edges. The limiting law isFor each fixed finite set of edges the restriction is unambiguous once $k$ is large. The standard exhaustion theorem guarantees that both limits exist and do not depend on the chosen connected exhaustion. Both are random spanning forests; despite their names, they need not be connected.