Electron-phonon interaction in two dimensions: The case for strong coupling in high-$T_c$ superconductors

A particular model interaction between electrons and acoustic phonons is found to cause electrons to become weakly "self-trapped" for an arbitrary filling factor in $d=2\mathrm{}$, but not in $d=1\mathrm{}$ nor, for different reasons, in $d\ge 3$. To the extent that electron kinetic energy is reduced in $d=2\mathrm{}$, the electronic interactions become proportionally more effective. Among phonon-mediated interactions we identify a static anisotropic interaction, compatible with observed ordering of oxygen vacancies along privileged axes in $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$, while other mechanisms encourage electron pairing and superconductivity.