Vortex waistlines and long range fluctuations

We examine the manner in which a linear potential results from fluctuations due to vortices linked with the Wilson loop. Our discussion is based on exact relations and inequalities between the Wilson loop and the vortex and electric flux order parameters. We show that, contrary to the customary naive picture, only vortex fluctuations of a thickness of the order of the spatial linear size of the loop are capable of producing a strictly linear potential. An effective theory of these long range fluctuations emerges naturally in the form of a strongly coupled $Z\left(N\right)\mathrm{}$ lattice gauge theory. We also point out that dynamical fermions introduced in this medium undergo chiral symmetry breaking.