  We investigate the statistical properties of interfering directed paths in
disordered media. At long distance, the average sign of the sum over paths may
tend to zero (sign-disordered) or remain finite (sign-ordered) depending on
dimensionality and the concentration of negative scattering sites $x$. We show
that in two dimensions the sign-ordered phase is unstable even for arbitrarily
small $x$ by identifying rare destabilizing events. In three dimensions, we
present strong evidence that there is a sign phase transition at a finite $x_c
> 0$. These results have consequences for several different physical systems.
In 2D insulators at low temperature, the variable range hopping
magnetoresistance is always negative, while in 3D, it changes sign at the point
of the sign phase transition. We also show that in the sign-disordered regime a
small magnetic field may enhance superconductivity in a random system of D-wave
superconducting grains embedded into a metallic matrix. Finally, the existence
of the sign phase transition in 3D implies new features in the spin glass phase
diagram at high temperature.
