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Abstract
We give a new proof of a result of Rick Kenyon that the probability that an edge in the middle of an $n \times n$ square is used in a loop-erased walk connecting opposite sides is of order $n^{-3/4}$. We, in fact, improve the result by showing that this estimate is correct up to multiplicative constants.
Highlights
Loop-erased random walk is a process obtained by erasing loops from simple random walk
One of the main motivations for doing the estimates in this paper is to show that the looperased random walk converges to Schramm-Loewner evolution with parameter 2 (SLE2) in the natural parametrization [5, 7]
The loop measure is the measure on unrooted loops induced by the above measure on rooted loops. (This representation of the measure on unrooted loops focuses on the rooted representative with root as far from the origin as possible.) The Brownian loop measure is the scaling limit of the random walk loop measure in a sense made precise in [8]
Summary
Loop-erased random walk is a process obtained by erasing loops from simple random walk. Where ≈ indicates that the logarithms of both sides are asymptotic His proof used the relationship between loop-erased walks and two other models, dimers and uniform spanning trees. A combinatorial identity is proved which writes the left-hand side of (1) in terms of simple random walk quantities. The proof is self-contained (other than some estimates for simple random walk) it does use a key idea from Kenyon’s work as discussed in [2, Section 5.7]. After the main result is proved, we generalize the combinatorial identity to random walk starting at any two points on ∂An. The argument is essentially the same, and one can compute the dependence of the probability on the starting points. This dependence on the boundary point is explained in terms of the scaling limit of loop-erased random walk, the Schramm-Loewner evolution with parameter SLE2. Up-to-constant estimates for the loop-erased walk probability can be viewed as a step in the program to establish this result
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