On the relation between the width of the flux tube and $T_c^{-1}$ in   lattice gauge theories
  Within the framework of a quantum flux tube model for the interquark potential it is possible to predict that in (2+1) dimensions the space-like string tension must increase with the temperature in the deconfined phase and that the thickness of the flux tube must coincide with the inverse of the deconfinement temperature. Both these predictions are in good agreement with some recent numerical simulations of SU(2) and $Z_2$ gauge models. 
M. Caselle, F. Gliozzi and S. Vinti