Geometrical Properties of Loops and Cluster Boundaries

We discuss how the statistical properties of the area and radius of gyration of single self-avoiding loops, and of Ising and percolation cluster boundaries, may be calculated using ideas of two-dimensional field theory. For cluster boundaries, we show that almost all loops have area $C\ln L+O(1)$, where $L$ is the size of the system, and $C$ is a calculable constant. We also compute the universal ratios $\langle A\rangle_\ell/\langle R^2\rangle_\ell$ of the area to the squared radius of gyration of loops of a given large perimeter $\ell$.