  Using neutron pair distribution function (PDF) analysis over the temperature
range from 1000 K to 15 K, we demonstrate the existence of local polarization
and the formation of medium-range, polar nanoregions (PNRs) with local
rhombohedral order in a prototypical relaxor ferroelectric
Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_3$. We estimate the volume fraction of the PNRs as a
function of temperature and show that this fraction steadily increases from 0 %
to a maximum of $\sim$ 30% as the temperature decreases from 650 K to 15 K.
Below T$\sim$200 K the PNRs start to overlap as their volume fraction reaches
the percolation threshold. We propose that percolating PNRs and their
concomitant overlap play a significant role in the relaxor behavior of
Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_3$.
