Growth exponent for loop-erased random walk in three dimensions

Abstract

Let $M_{n}$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^{3}$ run until its first exit from a ball of radius $n$. In the paper, we will show the existence of the growth exponent, that is, we show that there exists $\beta>0$ such that \begin{equation*}\lim_{n\to\infty}\frac{\log E(M_{n})}{\log n}=\beta.\end{equation*}