  Confining strings in 4D are effective, thick strings describing the
confinement phase of compact U(1) and, possibly, also non-Abelian gauge fields.
We show that these strings are dual to the gauge fields, inasmuch as their
perturbative regime corresponds to the strong coupling (large e) regime of the
gauge theory. In this regime they describe smooth surfaces with long-range
correlations and Hausdorff dimension two. For lower couplings e and monopole
fugacities z, a phase transition takes place, beyond which the smooth string
picture is lost. On the critical line intrinsic distances on the surface
diverge and correlators vanish, indicating that world-sheets become fractal.
