Universal amplitude in the sizes of rings in two dimensions

Abstract

For self-avoiding rings of N steps in two dimensions, the limiting value as N to infinity of the combination Np_N(R²)_Nx_c^N (where p_N is the number of distinct rings, (R²)_N is their mean square radius of gyration, and x_c is the critical fugacity) is equal to a calculable lattice-dependent number times a universal amplitude. This latter quantity is calculated exactly using methods of conformal invariance. The value is in good agreement with the results of enumeration studies.