  Let $M_{n}$ be the length (number of steps) of the loop-erasure of a simple
random walk up to the first exit from a ball of radius $n$ centered at its
starting point. It is shown in [18] that there exists $\beta \in (1,
\frac{5}{3}]$ such that $E (M_{n} )$ is of order $n^{\beta}$ in 3 dimensions.
In the present article, we show that the Hausdorff dimension of the scaling
limit of the loop-erased random walk in 3 dimensions is equal to $\beta$ almost
surely.

Probability