Mean area of self-avoiding loops

The mean area of two-dimensional unpressurized vesicles, or self-avoiding loops of fixed length N, behaves for large N as A₀{N}3/2$, while their mean square radius of gyration behaves as ${R}₀²N3/2. The amplitude ratio A₀/R₀² is computed exactly and found to equal 4{π}/5. The physics of the pressurized case, in both the inflated and collapsed phases, may be usefully related to that of a complex O(n) field theory coupled to a U(1) gauge field, in the limit n{→}0.