Peculiar scaling of self-avoiding walk contacts

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 --> infinity they are strictly finite in number but their radius of gyration R-c is power law distributed proportional to R-c(-tau), where tau > 1 is a novel exponent characterizing universal behavior. A continuum of diverging length scales is associated with the R-c distribution. A possibly superuniversal tau= 2 is also expected for the contacts of a self-avoiding or random walk with a confining wall.