The probability that planar loop-erased random walk uses a given edge

TL;DR

This paper provides a new proof and improves the estimate for the probability that a specific edge is used in a loop-erased random walk crossing a square, confirming it is proportional to n^{-3/4} within constant factors.

Contribution

The authors offer a new proof of Kenyon's result and refine the estimate to be accurate up to multiplicative constants.

Findings

Confirmed the probability is proportional to n^{-3/4} within constant factors.

Provided a new proof technique for the probability estimate.

Improved the precision of the known asymptotic probability estimate.

Abstract

We give a new proof of a result of Rick Kenyon that the probability that an edge in the middle of an n x n square is used in a loop-erased walk connecting opposites sides is of order n^{-3/4}. We, in fact, improve the result by showing that this estimate is correct up to multiplicative constants.