Mean Area of Self-Avoiding Loops

TL;DR

This paper investigates the asymptotic behavior of the mean area and radius of gyration of large self-avoiding loops in two dimensions, providing exact amplitude ratios and relating the physics to complex field theories.

Contribution

It provides an exact calculation of the amplitude ratio between mean area and mean square radius of gyration for self-avoiding loops and links the physics to complex O(n) field theory with gauge coupling.

Findings

Mean area scales as N^{3/2} with amplitude A_0

Mean square radius of gyration scales as N^{3/2} with amplitude R_0

Amplitude ratio A_0/R_0^2 equals 4π/5

Abstract

The mean area of two-dimensional unpressurised vesicles, or self-avoiding loops of fixed length $N$, behaves for large $N$ as $A_0 N^{3/2}$, while their mean square radius of gyration behaves as $R^2_0 N^{3/2}$. The amplitude ratio $A_0/R_0^2$ is computed exactly and found to equal $4\pi/5$. The physics of the pressurised case, both in 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 $n \to 0$.