  We develop a simple diagrammatic low-energy boson exchange strong-coupling
model for an antiferromagnetically correlated polaronic metal. Important
low-temperature features of such a model are (i) a large polaronic mass
enhancement $Z$ and a strongly narrowed, low-energy, near-Fermi-level van Hove
resonance peak in the interacting density of states at temperatures $T$ well
below the lattice tunneling energy scale $\Omega_t$; (ii) a rapid thermal
suppression of $Z$ and of the van Hove resonance which occurs when the
temperature $T$ becomes comparable to the lattice tunneling excitation energy
scale $\Omega_t$; and (iii) strong isotopic mass dependence of the electronic
mass enhancement $Z$ at temperatures $T$ well below $\Omega_t$. Surprisingly,
in spite of the large low-$T$ mass enhancment, the anharmonic lattice
fluctuations in the high temperature regime $T\gsim\Omega_t$, give rise only to
a moderate, essentially $T$-independent quasi-particle damping. We suggest
possible low temperature isotope experiments to explore polaronic fluctuations
in the cuprates superconductors.
