  The distribution of thermally excited vortex loops near a superfluid phase
transition is calculated from a renormalized theory. The number density of
loops with a given perimeter is found to change from exponential decay with
increasing perimeter to algebraic decay as T_c is approached, in agreement with
recent simulations of both cosmic strings and high-T_c superconductors.
Predictions of the value of the exponent of the algebraic decay at T_c and of
critical behavior in the vortex density are confirmed by the simulations,
giving strong support to the vortex-folding model proposed by Shenoy.
