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).