Vortex waistlines and vortex gas

A large body of work over the last few years [1] supports the picture of long thick vortices as the most important long range degrees of freedom for confinement. In this note we wish to examine more closely the manner in which a linear potential arises from vortex fluctuations. One customarily assumes a picture of randomly distributed vortices of certain thickness and basically arbitrary length. Furthermore, in the presence of a Wilson loop, one assumes that the free energy cost and hence the probability for a vortex to pierce anywhere inside the area enclosed by the loop is constant for all loop sizes. Summing then independently over all vortices linking with the loop one gets area law behavior for the loop expectation: