  Relaxor ferroelectrics are complex oxide materials which are rather unique to
study the effects of compositional disorder on phase transitions. Here, we
study the effects of quenched cubic random electric fields on the lattice
instabilities that lead to a ferroelectric transition and show that, within a
microscopic model and a statistical mechanical solution, even weak
compositional disorder can prohibit the development of long-range order and
that a random field state with anisotropic and power-law correlations of
polarization emerges from the combined effect of their characteristic dipole
forces and their inherent charge disorder. We compare and reproduce several key
experimental observations in the well- studied relaxor
PbMg$_{1/3}$Nb$_{2/3}$O$_3$-PbTiO$_3$.
