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For finite $G$, choose a root $r$ and an ordering of the other graph vertices. Begin with a tree consisting only of $r$. At each stage take the first graph vertex not already in the tree, run an independent simple random walk from it until its first hit of the existing tree, erase the loops chronologically, and add the resulting graph path. In chronological loop erasure, revisiting a graph vertex already in the current list deletes all list entries after that graph vertex; otherwise append the new graph vertex. The retained final graph path is simple, with new internal graph vertices and exactly one endpoint in the old tree. Adding it therefore preserves connectedness and creates no graph cycle. Graph vertices already included are skipped. On a finite connected graph each walk hits the existing tree almost surely; after finitely many attachments all graph vertices are present.

The Wilson algorithm theorem states that this tree is a uniform spanning tree, independently of the chosen root and order. The required hypotheses here are an undirected connected finite graph and transition probabilities proportional to conductance; unit conductances give simple random walk and the uniform law. This is the sampling result being described, not a claim that arbitrary walk transition probabilities would produce a uniform tree.

For an infinite connected locally finite recurrent graph, choose a fixed finite root $r$ and enumerate all graph vertices. A connected locally finite graph is countable, since its finite-radius balls are finite. Run the same algorithm successively. Recurrence and irreducibility imply that a walk from any graph vertex hits $r$ almost surely, hence hits the current tree. Every attached graph path is therefore finite. The countable intersection of these probability-one events has probability one, and the increasing union is an acyclic connected subgraph containing all graph vertices: each graph vertex receives a finite route to $r$.

This gives the uniform spanning tree of a recurrent infinite graph. It is not a uniform counting measure on all infinite spanning trees. Equivalently, its joint probability distributions on finite collections of edges are the common limits of uniform spanning trees along any graph exhaustion, using the approximations defining the free uniform spanning forest and the wired uniform spanning forest. To check that equivalence, fix a finite edge set $F$ and run the chosen enumeration through a finite prefix containing all its endpoints. Once those graph vertices are in the tree, later attachments cannot add an edge with both endpoints already included, so membership of $F$ is settled permanently. Only finitely many random walks have been used; each has a finite trajectory almost surely. All their visited graph vertices and all their graph neighbours lie in a sufficiently large exhaustion set. Couple finite-volume and infinite-volume walks with the same steps there. Whether the exterior is deleted or wired to one boundary graph vertex, these initial trajectories and their loop erasures then agree. This proves convergence of the probabilities of cylinder sets determined by those edges.

The limiting law is unchanged by the root and ordering because the laws on finite graphs have this invariance. The same coupling shows that changing the exhaustion also leaves the law unchanged. ThusThe result is a single spanning tree because the graph is recurrent. On a transient graph, infinite-volume spanning-tree limits may instead be uniform spanning forests; recurrence cannot be omitted from this conclusion.

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.

On a tree, every finite connected induced subgraph is itself a tree, and its only spanning tree contains all its edges. Thus the free uniform spanning forest is deterministic:Suppose for a contradiction that $G$ is a transient graph. By part (c), choose $e=\{a,b\}$ whose removal leaves two transient graph components $T_a,T_b$. Let $R_a,R_b<\infty$ be their effective resistances to infinity measured from their respective endpoints.

In a wired finite exhaustion, the edge $e$ of resistance one is in parallel with the route from $a$ through $T_a$ to the wired boundary and back through $T_b$ to $b$. The latter route has resistance $R_{a,k}+R_{b,k}$, converging to $R_a+R_b$. Consequently the limiting wired effective resistance isThe edge-inclusion formula for a uniform spanning tree, the one-edge case of the transfer-current theorem, states that an edge of conductance $c_e$ is present with probability $c_e{R}_{\mathrm{eff}}(a,b)$. Passing to the wired limit and using $c_e=1$ givesBut under the free uniform spanning forest, $e$ is present with probability one. This contradicts $\mathrm{WSF}=\mathrm{FSF}$. Therefore the tree is recurrent. Combined with part (b), this characterizes equality of the free and wired uniform spanning forests on locally finite unweighted trees.

Let ${\mu }_n$ be the uniform spanning tree measure on the finite connected induced graph $G_n$. Uniform spanning-tree edge indicators are negatively associated: increasing events depending on disjoint edge sets have nonpositive covariance. Together with the spatial Markov property, this implies the free-boundary monotonicity under graph enlargement: for every increasing cylinder event $A$, ${\mu }_n(A)$ is eventually nonincreasing once $G_n$ contains all edges on which $A$ depends.

Define the free uniform spanning forest measure byfor increasing cylinder events. These limits determine a unique probability measure, independent of the exhaustion; equivalently, ${\mu }_n$ converges weakly to ${\mu }_F$ on the product space.

A uniform spanning forest is an infinite-volume limit of uniform spanning trees on finite graph approximations. Free and wired boundary identifications give the free uniform spanning forest and wired uniform spanning forest, respectively.