  The nearest neighbor contacts between the two halves of an N-site lattice
self-avoiding walk offer an unusual example of scaling random geometry: for N
going to infinity they are strictly finite in number but their radius of
gyration Rc is power law distributed, ~ Rc^{-\tau}, where \tau>1 is a novel
exponent characterizing universal behavior. A continuum of diverging lengths
scales is associated to the Rc distribution. A possibly super-universal \tau=2
is also expected for the contacts of a self-avoiding or random walk with a
confining wall.
