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 $N\to \infty$ they are strictly finite in number but their radius of gyration $R_c$ is power law distributed $\propto R_c^{-\tau }$, where $\tau >1\mathrm{}$ is a novel exponent characterizing universal behavior. A continuum of diverging length scales is associated with the $R_c$ distribution. A possibly superuniversal $\tau =2\mathrm{}$ is also expected for the contacts of a self-avoiding or random walk with a confining wall.

• Received 13 March 2001

DOI:https://doi.org/10.1103/PhysRevLett.87.070602