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Conformal invariance of planar loop-erased random walks and uniform spanning trees. (English) Zbl 1126.82011
Summary: This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $$D\subsetneqq\mathbb C$$ is equal to the radial $$\text{SLE}_2$$ path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that $$\partial D$$ is a $$C^1$$-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $$A\subset\partial D$$, is the chordal $$\text{SLE}_8$$ path in $$\overline D$$ joining the endpoints of $$A$$. A by-product of this result is that $$\text{SLE}_8$$ is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

##### MSC:
 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 60G50 Sums of independent random variables; random walks
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