The growth exponent for planar loop-erased random walk

TL;DR

This paper provides a new proof that the growth exponent for planar loop-erased random walks is 5/4, utilizing convergence to Schramm-Loewner evolution, applicable to various two-dimensional lattices.

Contribution

It introduces a novel proof of the growth exponent for 2D LERW using SLE convergence, extending validity beyond specific lattice types.

Findings

The growth exponent for 2D LERW is 5/4.

The proof applies to any irreducible bounded symmetric random walk on 2D lattices.

Convergence to SLE(2) is central to the proof.

Abstract

We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two-dimensional discrete lattice.