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One precise relation is to radial SLE with parameter $2$. Its orientation is from the boundary point $1$ to the interior target $0$. With conformal-radius time it is defined byThere is a coupling of the conditioned Brownian path $B$ with this simple radial curve $\eta$ such that, for each initial curve segment, the first point at which $B$ encounters it is its tip. More precisely, if $s_t=\inf \{s:B_s\in \eta [0,t]\}$, then $B_{s_t}=\eta (t)$. Reversing the Brownian path turns these first-hit times into the last-visit times that characterize loop-erasure of planar Brownian motion. ThusThis is an existence statement for a coupling, not a pointwise equality of the two processes or a claim that a naive finite-loop deletion algorithm applies to Brownian motion.

Here is the martingale mechanism behind the coupling. Put $a_t=e^{i{\theta }_t}$ andThis is the positive slit-domain Poisson kernel at the tip, normalized by $Q_t(0)=1$. Writing $q_t=g_t(z)/a_t$, the radial equation givesThe drift of $(1+q_t)/(1-q_t)$ is $(2-\kappa )q_t(1+q_t)/(1-q_t{)}^3$, so $Q_t(z)$ is a local martingale at $\kappa =2$.

Start independent stopped $B$ and radial SLE2, and weight their joint laws, before meeting, byFor fixed $s$ this is an SLE local martingale. For fixed $t$ it is a Brownian-transform local martingale, because ${\mathcal{L}}^h(Q_t/h)=(2h{)}^{-1}\Delta Q_t=0$. Since $L(0,t)=L(s,0)=1$, localized two-parameter weighting preserves both original marginals. For a fixed slit, the changed Brownian law is the transform by $Q_t$, so it hits that slit at the pole, its tip. Compatible stopped couplings, followed by exhaustion, give the first-hit-at-tip property. This completion is the substantive continuum coupling step. Discrete loop-erased random walks and their radial-SLE2 limit give a parallel approximation picture.

For example, in this coupling avoidance by $B$ implies avoidance by $\eta$, since the radial curve is contained in the Brownian trace. The probability from part (b) therefore supplies a lower bound $|{\varphi }^{\prime }(1)|\le \mathbb{P}(\eta \subset U)$. The two avoidance laws are not identical: unweighted simple radial ${\mathrm{SLE}}_2$ does not have the Brownian radial-restriction property. This is a way to transfer path-containment information while retaining the distinct geometry and orientation of the two random curves.

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.

For radial SLE at parameter two, the normalized slit-domain Poisson kernel at the growing tip is a local martingale. This observable provides the coupling to loop-erasure of planar Brownian motion, oriented from the boundary point to the interior target. Its discrete counterpart is a loop-erased random walk.

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.