Geometrical Properties of Loops and Cluster Boundaries

Abstract

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 $⟨A{⟩}_{\ell }/⟨R^2{⟩}_{\ell }$ of the area to the squared radius of gyration of loops of a given large perimeter $\ell$.

13 pages, 2 figures. Two lectures presented at 1994 Les Houches Summer School ``Fluctuating Geometries in Statistical Mechanics and Field Theory'' (also available at http://xxx.lanl.gov/lh94/ )