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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.

First take $x\ne y$. Sample a uniform spanning tree using Wilson's algorithm rooted at $y$, with $x$ processed first. The first walk runs from $x$ until its first hit of $y$, so its loop erasure is exactly the unique $x$-to-$y$ graph path of the resulting tree. Later attachments cannot alter that graph path. This is the uniform-spanning-tree path representation of loop-erased random walk.

Instead root Wilson's algorithm at $x$ and process $y$ first. The resulting tree still has the same uniform law, while its first attachment has the law of loop-erased random walk from $y$ to $x$. A tree's graph path between two fixed graph vertices has the same unoriented edge set in both directions. Pushing the common tree law through this graph path map therefore givesHere $\mathcal{E}$ means the set of unoriented edges. In fact, reversing the complete graph vertex sequence from the first loop-erased walk has the law of the second, giving reversibility of loop-erased random walk. If $x=y$ and first hitting allows time zero, both graph paths are empty and the equality is immediate.

This is a distributional assertion, not a pathwise identity between chronological loop erasure and reversal. For example, the valid walk trajectory $x,a,b,a,c,b,y$ in a graph containing the triangle on $a,b,c$ and the edges $xa,by$ erases forward to $x,a,c,b,y$. Erasing the reversed trajectory and then reversing the result gives $x,a,b,y$, a different graph path. Wilson's root-independent tree law is what proves the stated equality of probability distributions.

Fix a root $r$. On a recurrent graph, a simple random walk started at any vertex hits $r$ almost surely. Run Wilson's algorithm with root $r$: successively attach the loop-erased random walk from each new starting vertex, stopped when it hits the existing tree. Every walk terminates almost surely, because that tree already contains $r$.

To compare the two finite approximations, fix a finite set of edges $K$ and put all their endpoints first in the ordering of starting vertices. In the infinite graph, these finitely many Wilson's algorithm walks have finite lengths almost surely. Their visited vertices, together with their neighbors, are therefore contained in $V_k$ for all sufficiently large $k$ on each realization.

Couple the walks in $G[V_k]$, in $G_k^{\mathrm{w}}$, and in $G$ using the same neighbor choices while they are in the interior of $V_k$. On the event just described, they encounter neither boundary, so the walks, their chronological loop erasures, and the resulting partial trees agree. Once every endpoint of $K$ has entered the partial tree, later walks cannot add any edge of $K$: they stop on their first hit of the existing tree. Thus the indicators of membership for all edges of $K$ agree in the free and wired uniform spanning trees, with probability tending to one.

The two limiting measures agree on every finite-edge event, and thereforeThis argument also shows that the common uniform spanning forest is a single spanning tree on a recurrent graph. The equality refers to probability measures, not to independently sampled forests being identical.

Run Wilson algorithm with root $a$, beginning with a simple random walk from $x_1$. Its chronological loop erasure is added as the first branch, so the resulting tree contains a path from $x_1$ to $a$ with exactly the law of the stated loop-erased random walk.

For completeness, the algorithm produces a uniform spanning tree. For any prescribed ordered collection of loop-erased branches, the random-walk decomposition gives a product of local transition factors and diagonal killed Green functions. The transition factors depend only on the resulting oriented tree, while the product of Green functions is independent of the vertex order by Question 3. Thus every spanning tree receives the same probability. Since a tree has a unique simple path between two vertices, its $x_1$-to-$a$ path has the claimed law.

Take an increasing exhaustion $G_n$ of $G$ by finite connected vertex sets, identify every vertex outside $G_n$ to one boundary vertex, and choose a uniform spanning tree of the resulting finite wired graph. The weak limit as $n\to \infty$ is the wired uniform spanning forest of $G$.

For Wilson algorithm rooted at infinity, enumerate $V$ as $v_1,v_2,\dots$. Run a simple random walk from $v_1$ forever and add its chronological loop erasure. Transience makes the infinite loop erasure well-defined. Next, from the first vertex not already in the forest, run an independent random walk until it hits the existing forest; if it never hits, run it forever. Add its loop erasure and continue through the enumeration. Wilson's theorem rooted at infinity says that the resulting forest has the wired uniform spanning forest law.

On a finite connected unweighted graph, the graph path between $x$ and $y$ in a uniform spanning tree has the law of loop-erased random walk from $x$ to $y$. Root Wilson's algorithm at $y$ and process $x$ first: the first attached graph path is exactly that loop erasure, and later attachments cannot change it. Changing the root to $x$ proves reversibility of loop-erased random walk, even though chronological loop erasure does not commute with reversing each individual random-walk trajectory.

Wilson's algorithm rooted at infinity builds the wired uniform spanning forest on a transient graph by successively adding loop-erased random walks, run forever when they never hit the forest already constructed.