For a smooth curve, the natural paraemtrization
is parametrization by arc length. What is the analogue
for a random curve of fractal dimension d? Typically,
such curves have Hausdorff dmeasure 0. It turns out
that a different quantity, Minkowski content, is the
right thing.
I will discuss results of this type for the Schramm-Loewner
evolution --- both how to prove the content is well-defined
(work with M. Rezaei) and how it relates to the scaling
limit of the loop-erased random walk (work with F. Viklund
and C. Benes).